Chain rule - Wikipedia

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class="vector-toc-numb">4.3</span> <span>Derivatives of inverse functions</span> </div> </a> <ul id="toc-Derivatives_of_inverse_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Back_propagation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Back_propagation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Back propagation</span> </div> </a> <ul id="toc-Back_propagation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Higher_derivatives" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Higher derivatives</span> </div> </a> <ul id="toc-Higher_derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proofs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Proofs</span> </div> </a> <button aria-controls="toc-Proofs-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Proofs subsection</span> </button> <ul id="toc-Proofs-sublist" class="vector-toc-list"> <li id="toc-First_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>First proof</span> </div> </a> <ul id="toc-First_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Second proof</span> </div> </a> <ul id="toc-Second_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Third_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Third_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Third proof</span> </div> </a> <ul id="toc-Third_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_via_infinitesimals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_via_infinitesimals"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Proof via infinitesimals</span> </div> </a> <ul id="toc-Proof_via_infinitesimals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Multivariable_case" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multivariable_case"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Multivariable case</span> </div> </a> <button aria-controls="toc-Multivariable_case-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Multivariable case subsection</span> </button> <ul id="toc-Multivariable_case-sublist" class="vector-toc-list"> <li id="toc-Case_of_scalar-valued_functions_with_multiple_inputs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Case_of_scalar-valued_functions_with_multiple_inputs"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Case of scalar-valued functions with multiple inputs</span> </div> </a> <ul id="toc-Case_of_scalar-valued_functions_with_multiple_inputs-sublist" class="vector-toc-list"> <li id="toc-Example:_arithmetic_operations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example:_arithmetic_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1.1</span> <span>Example: arithmetic operations</span> </div> </a> <ul id="toc-Example:_arithmetic_operations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_rule:_Vector-valued_functions_with_multiple_inputs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_rule:_Vector-valued_functions_with_multiple_inputs"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>General rule: Vector-valued functions with multiple inputs</span> </div> </a> <ul id="toc-General_rule:_Vector-valued_functions_with_multiple_inputs-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_derivatives_of_multivariable_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Higher_derivatives_of_multivariable_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2.2</span> <span>Higher derivatives of multivariable functions</span> </div> </a> <ul id="toc-Higher_derivatives_of_multivariable_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Further_generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further generalizations</span> </div> </a> <ul id="toc-Further_generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Chain rule</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 43 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-43" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">43 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Kettingre%C3%ABl" title="Kettingreël – Afrikaans" lang="af" hreflang="af" data-title="Kettingreël" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%B9%D8%AF%D8%A9_%D8%A7%D9%84%D8%B3%D9%84%D8%B3%D9%84%D8%A9" title="قاعدة السلسلة – Arabic" lang="ar" hreflang="ar" data-title="قاعدة السلسلة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Li%C3%A2n-s%C3%B3-lu%CC%8Dt" title="Liân-só-lu̍t – Minnan" lang="nan" hreflang="nan" data-title="Liân-só-lu̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%92%D0%B5%D1%80%D0%B8%D0%B6%D0%BD%D0%BE_%D0%BF%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE" title="Верижно правило – Bulgarian" lang="bg" hreflang="bg" data-title="Верижно правило" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Pravilo_derivacije_slo%C5%BEene_funkcije" title="Pravilo derivacije složene funkcije – Bosnian" lang="bs" hreflang="bs" data-title="Pravilo derivacije složene funkcije" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Regla_de_la_cadena" title="Regla de la cadena – Catalan" lang="ca" hreflang="ca" data-title="Regla de la cadena" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D1%83%D1%82%D0%BB%C4%83_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BD_%D1%82%D1%83%D1%85%D1%81%D0%B0%D1%82%C4%83%D1%80%D0%B0%D0%BD%D0%BD%D0%B5_%D1%82%D1%83%D0%BF%D0%B0%D1%81%D1%81%D0%B8" title="Хутлă функцин тухсатăранне тупасси – Chuvash" lang="cv" hreflang="cv" data-title="Хутлă функцин тухсатăранне тупасси" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C5%98et%C3%ADzkov%C3%A9_pravidlo" title="Řetízkové pravidlo – Czech" lang="cs" hreflang="cs" data-title="Řetízkové pravidlo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kettenregel" title="Kettenregel – German" lang="de" hreflang="de" data-title="Kettenregel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Regla_de_la_cadena" title="Regla de la cadena – Spanish" lang="es" hreflang="es" data-title="Regla de la cadena" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Deriva%C4%B5o_de_funkcia_kompona%C4%B5o" title="Derivaĵo de funkcia komponaĵo – Esperanto" lang="eo" hreflang="eo" data-title="Derivaĵo de funkcia komponaĵo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Katearen_erregela" title="Katearen erregela – Basque" lang="eu" hreflang="eu" data-title="Katearen erregela" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D8%B9%D8%AF%D9%87_%D8%B2%D9%86%D8%AC%DB%8C%D8%B1%D9%87%E2%80%8C%D8%A7%DB%8C" title="قاعده زنجیره‌ای – Persian" lang="fa" hreflang="fa" data-title="قاعده زنجیره‌ای" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_d%C3%A9rivation_des_fonctions_compos%C3%A9es" title="Théorème de dérivation des fonctions composées – French" lang="fr" hreflang="fr" data-title="Théorème de dérivation des fonctions composées" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Regra_da_cadea" title="Regra da cadea – Galician" lang="gl" hreflang="gl" data-title="Regra da cadea" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%B0%EC%87%84_%EB%B2%95%EC%B9%99" title="연쇄 법칙 – Korean" lang="ko" hreflang="ko" data-title="연쇄 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D6%80%D5%A4_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1%D5%B5%D5%AB_%D5%A4%D5%AB%D6%86%D5%A5%D6%80%D5%A5%D5%B6%D6%81%D5%B8%D6%82%D5%B4" title="Բարդ ֆունկցիայի դիֆերենցում – Armenian" lang="hy" hreflang="hy" data-title="Բարդ ֆունկցիայի դիֆերենցում" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B6%E0%A5%83%E0%A4%82%E0%A4%96%E0%A4%B2%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="शृंखला नियम – Hindi" lang="hi" hreflang="hi" data-title="शृंखला नियम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kaidah_rantai" title="Kaidah rantai – Indonesian" lang="id" hreflang="id" data-title="Kaidah rantai" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ke%C3%B0juregla" title="Keðjuregla – Icelandic" lang="is" hreflang="is" data-title="Keðjuregla" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Regola_della_catena" title="Regola della catena – Italian" lang="it" hreflang="it" data-title="Regola della catena" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A9%D7%A8%D7%A9%D7%A8%D7%AA" title="כלל השרשרת – Hebrew" lang="he" hreflang="he" data-title="כלל השרשרת" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/L%C3%A1ncszab%C3%A1ly" title="Láncszabály – Hungarian" lang="hu" hreflang="hu" data-title="Láncszabály" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kettingregel" title="Kettingregel – Dutch" lang="nl" hreflang="nl" data-title="Kettingregel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%A3%E9%8E%96%E5%BE%8B" title="連鎖律 – Japanese" lang="ja" hreflang="ja" data-title="連鎖律" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kjerneregelen" title="Kjerneregelen – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kjerneregelen" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/R%C3%A9gola_dla_caden-a" title="Régola dla caden-a – Piedmontese" lang="pms" hreflang="pms" data-title="Régola dla caden-a" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Regu%C5%82a_%C5%82a%C5%84cuchowa" title="Reguła łańcuchowa – Polish" lang="pl" hreflang="pl" data-title="Reguła łańcuchowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Regra_da_cadeia" title="Regra da cadeia – Portuguese" lang="pt" hreflang="pt" data-title="Regra da cadeia" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Derivarea_func%C8%9Biilor_compuse" title="Derivarea funcțiilor compuse – Romanian" lang="ro" hreflang="ro" data-title="Derivarea funcțiilor compuse" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D1%81%D0%BB%D0%BE%D0%B6%D0%BD%D0%BE%D0%B9_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Дифференцирование сложной функции – Russian" lang="ru" hreflang="ru" data-title="Дифференцирование сложной функции" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Chain_rule" title="Chain rule – Simple English" lang="en-simple" hreflang="en-simple" data-title="Chain rule" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Odvod_kompozituma" title="Odvod kompozituma – Slovenian" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">For derivatives of composed functions</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about composed derivatives in calculus. For the probability theory concept, see <a href="/wiki/Chain_rule_(probability)" title="Chain rule (probability)">Chain rule (probability)</a>. For other uses, see <a href="/wiki/Chain_rule_(disambiguation)" class="mw-disambig" title="Chain rule (disambiguation)">Chain rule (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}" /></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a class="mw-selflink selflink">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><br /><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Fractional_calculus" title="Fractional calculus">Fractional</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Miscellanea</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History</a></li> <li><a href="/wiki/Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li> <li><a href="/wiki/List_of_calculus_topics" title="List of calculus topics">List of topics</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus" title="Template:Calculus"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus" title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Calculus" title="Calculus">calculus</a>, the <b>chain rule</b> is a <a href="/wiki/Formula" title="Formula">formula</a> that expresses the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the <a href="/wiki/Function_composition" title="Function composition">composition</a> of two <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable functions</a> <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> in terms of the derivatives of <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span>. More precisely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=f\circ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14345f3938c66b1fde08e18d8b301a59c5775a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.027ex; height:2.509ex;" alt="{\displaystyle h=f\circ g}" /></span> is the function such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=f(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=f(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ac4cb5a6518f4fbf566c6b0d2f0af5d6551f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.919ex; height:2.843ex;" alt="{\displaystyle h(x)=f(g(x))}" /></span> for every <span class="texhtml mvar" style="font-style:italic;">x</span>, then the chain rule is, in <a href="/wiki/Lagrange%27s_notation" class="mw-redirect" title="Lagrange's notation">Lagrange's notation</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h'(x)=f'(g(x))g'(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'(x)=f'(g(x))g'(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c055a07905ad198e6ba0022fc32736cfd23daf4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.919ex; height:3.009ex;" alt="{\displaystyle h'(x)=f'(g(x))g'(x).}" /></span> or, equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767d917fb4f1e4bb20a60cebc2df701874d44e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.558ex; height:3.009ex;" alt="{\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}" /></span> </p><p>The chain rule may also be expressed in <a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a>. If a variable <span class="texhtml mvar" style="font-style:italic;">z</span> depends on the variable <span class="texhtml mvar" style="font-style:italic;">y</span>, which itself depends on the variable <span class="texhtml mvar" style="font-style:italic;">x</span> (that is, <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">z</span> are <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variables</a>), then <span class="texhtml mvar" style="font-style:italic;">z</span> depends on <span class="texhtml mvar" style="font-style:italic;">x</span> as well, via the intermediate variable <span class="texhtml mvar" style="font-style:italic;">y</span>. In this case, the chain rule is expressed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a610aa8446be002e2e30d7121f6a87273d4caa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.395ex; height:5.843ex;" alt="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋅</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ba9a37e86e54a62beb226f2949a840deff8765" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.95ex; height:6.343ex;" alt="{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}" /></span> for indicating at which points the derivatives have to be evaluated. </p><p>In <a href="/wiki/Integral" title="Integral">integration</a>, the counterpart to the chain rule is the <a href="/wiki/Substitution_rule" class="mw-redirect" title="Substitution rule">substitution rule</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Intuitive_explanation">Intuitive explanation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=1" title="Edit section: Intuitive explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Intuitively, the chain rule states that knowing the instantaneous rate of change of <span class="texhtml"><i>z</i></span> relative to <span class="texhtml"><i>y</i></span> and that of <span class="texhtml"><i>y</i></span> relative to <span class="texhtml"><i>x</i></span> allows one to calculate the instantaneous rate of change of <span class="texhtml"><i>z</i></span> relative to <span class="texhtml"><i>x</i></span> as the product of the two rates of change. </p><p>As put by <a href="/wiki/George_F._Simmons" title="George F. Simmons">George F. Simmons</a>: "If a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man."<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The relationship between this example and the chain rule is as follows. Let <span class="texhtml mvar" style="font-style:italic;">z</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> and <span class="texhtml mvar" style="font-style:italic;">x</span> be the (variable) positions of the car, the bicycle, and the walking man, respectively. The rate of change of relative positions of the car and the bicycle is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dz}{dy}}=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {dz}{dy}}=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c0a6d7a5d2d56a62796839af72d517b6023910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:7.421ex; height:4.176ex;" alt="{\textstyle {\frac {dz}{dy}}=2.}" /></span> Similarly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dy}{dx}}=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {dy}{dx}}=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3bcfe19742de78e60ec58cc82380def6132fc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.544ex; height:4.176ex;" alt="{\textstyle {\frac {dy}{dx}}=4.}" /></span> So, the rate of change of the relative positions of the car and the walking man is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=2\cdot 4=8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>=</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=2\cdot 4=8.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370629d3d6db86ea972c602a1e96a024ba912389" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.759ex; height:5.843ex;" alt="{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=2\cdot 4=8.}" /></span> </p><p>The rate of change of positions is the ratio of the speeds, and the speed is the derivative of the position with respect to the time; that is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dx}}={\frac {\frac {dz}{dt}}{\frac {dx}{dt}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dx}}={\frac {\frac {dz}{dt}}{\frac {dx}{dt}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/749ad4df58f2c327239d0a3d7c0a8915aae22691" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:10.599ex; height:8.176ex;" alt="{\displaystyle {\frac {dz}{dx}}={\frac {\frac {dz}{dt}}{\frac {dx}{dt}}},}" /></span> or, equivalently, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dz}{dt}}={\frac {dz}{dx}}\cdot {\frac {dx}{dt}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dz}{dt}}={\frac {dz}{dx}}\cdot {\frac {dx}{dt}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242782c9b45c4e1c5c2310b2d54fdae067d80a33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.328ex; height:5.509ex;" alt="{\displaystyle {\frac {dz}{dt}}={\frac {dz}{dx}}\cdot {\frac {dx}{dt}},}" /></span> which is also an application of the chain rule. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The chain rule seems to have first been used by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>. He used it to calculate the derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a+bz+cz^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>z</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a+bz+cz^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c0c92a5de65c0c1634238ed3c45da388d47e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.471ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a+bz+cz^{2}}}}" /></span> as the composite of the square root function and the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bz+cz^{2}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>z</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bz+cz^{2}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1befd87b2a49a7e475ba638d598d3e15578562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:12.148ex; height:2.843ex;" alt="{\displaystyle a+bz+cz^{2}\!}" /></span>. He first mentioned it in a 1676 memoir (with a sign error in the calculation).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The common notation of the chain rule is due to Leibniz.<sup id="cite_ref-OHR_3-0" class="reference"><a href="#cite_note-OHR-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Guillaume_de_l%27H%C3%B4pital" title="Guillaume de l'Hôpital">Guillaume de l'Hôpital</a> used the chain rule implicitly in his <i><a href="/wiki/Analyse_des_infiniment_petits" class="mw-redirect" title="Analyse des infiniment petits">Analyse des infiniment petits</a></i>. The chain rule does not appear in any of <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>'s analysis books, even though they were written over a hundred years after Leibniz's discovery.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2022)">citation needed</span></a></i>]</sup>. It is believed that the first "modern" version of the chain rule appears in Lagrange's 1797 <i>Théorie des fonctions analytiques</i>; it also appears in Cauchy's 1823 <i>Résumé des Leçons données a L’École Royale Polytechnique sur Le Calcul Infinitesimal</i>.<sup id="cite_ref-OHR_3-1" class="reference"><a href="#cite_note-OHR-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=3" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest form of the chain rule is for real-valued functions of one <a href="/wiki/Real_number" title="Real number">real</a> variable. It states that if <i><span class="texhtml mvar" style="font-style:italic;">g</span></i> is a function that is differentiable at a point <i><span class="texhtml mvar" style="font-style:italic;">c</span></i> (i.e. the derivative <span class="texhtml"><i>g</i>′(<i>c</i>)</span> exists) and <i><span class="texhtml mvar" style="font-style:italic;">f</span></i> is a function that is differentiable at <span class="texhtml"><i>g</i>(<i>c</i>)</span>, then the composite function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" /></span> is differentiable at <i><span class="texhtml mvar" style="font-style:italic;">c</span></i>, and the derivative is<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)'(c)=f'(g(c))\cdot g'(c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)'(c)=f'(g(c))\cdot g'(c).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8808665bd39eccfe83c378f9fe9260f8448b75e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.69ex; height:3.009ex;" alt="{\displaystyle (f\circ g)'(c)=f'(g(c))\cdot g'(c).}" /></span> The rule is sometimes abbreviated as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb55cd5448d4bed6da3b79283d92eec2ab9bb95d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.436ex; height:3.009ex;" alt="{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}" /></span> </p><p>If <span class="texhtml"><i>y</i> = <i>f</i>(<i>u</i>)</span> and <span class="texhtml"><i>u</i> = <i>g</i>(<i>x</i>)</span>, then this abbreviated form is written in <a href="/wiki/Leibniz_notation" class="mw-redirect" title="Leibniz notation">Leibniz notation</a> as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641aeadce2e1a4477095109d64d119700960d890" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.569ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}" /></span> </p><p>The points where the derivatives are evaluated may also be stated explicitly: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {dy}{dx}}\right|_{x=c}=\left.{\frac {dy}{du}}\right|_{u=g(c)}\cdot \left.{\frac {du}{dx}}\right|_{x=c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋅</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {dy}{dx}}\right|_{x=c}=\left.{\frac {dy}{du}}\right|_{u=g(c)}\cdot \left.{\frac {du}{dx}}\right|_{x=c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77480a909a9b77bc7502c83d3d0e69a8522118d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.067ex; height:6.343ex;" alt="{\displaystyle \left.{\frac {dy}{dx}}\right|_{x=c}=\left.{\frac {dy}{du}}\right|_{u=g(c)}\cdot \left.{\frac {du}{dx}}\right|_{x=c}.}" /></span> </p><p>Carrying the same reasoning further, given <i><span class="texhtml mvar" style="font-style:italic;">n</span></i> functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},\ldots ,f_{n}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},\ldots ,f_{n}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15f64424ef6f4cd26f9a9917eca3e39a6f5f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:9.729ex; height:2.509ex;" alt="{\displaystyle f_{1},\ldots ,f_{n}\!}" /></span> with the composite function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\circ (f_{2}\circ \cdots (f_{n-1}\circ f_{n}))\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∘</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}\circ (f_{2}\circ \cdots (f_{n-1}\circ f_{n}))\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3298b5b3aae4c5bd3532bc748f359d166f93e4da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:24.295ex; height:2.843ex;" alt="{\displaystyle f_{1}\circ (f_{2}\circ \cdots (f_{n-1}\circ f_{n}))\!}" /></span>, if each function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e151118d7cd691396e64e32ec2f73973a0304621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}\!}" /></span> is differentiable at its immediate input, then the composite function is also differentiable by the repeated application of Chain Rule, where the derivative is (in Leibniz's notation): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {df_{1}}{dx}}={\frac {df_{1}}{df_{2}}}{\frac {df_{2}}{df_{3}}}\cdots {\frac {df_{n}}{dx}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {df_{1}}{dx}}={\frac {df_{1}}{df_{2}}}{\frac {df_{2}}{df_{3}}}\cdots {\frac {df_{n}}{dx}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae35840534baa75024e266d5373e47c07e1faca1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.389ex; height:5.843ex;" alt="{\displaystyle {\frac {df_{1}}{dx}}={\frac {df_{1}}{df_{2}}}{\frac {df_{2}}{df_{3}}}\cdots {\frac {df_{n}}{dx}}.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chain_rule_en.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Chain_rule_en.png/350px-Chain_rule_en.png" decoding="async" width="350" height="465" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2d/Chain_rule_en.png 1.5x" data-file-width="420" data-file-height="558" /></a><figcaption>The chain rule in case of composites of more than two functions</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Composites_of_more_than_two_functions">Composites of more than two functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=5" title="Edit section: Composites of more than two functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The chain rule can be applied to composites of more than two functions. To take the derivative of a composite of more than two functions, notice that the composite of <span class="texhtml mvar" style="font-style:italic;">f</span>, <span class="texhtml mvar" style="font-style:italic;">g</span>, and <i><span class="texhtml mvar" style="font-style:italic;">h</span></i> (in that order) is the composite of <span class="texhtml mvar" style="font-style:italic;">f</span> with <span class="texhtml"><i>g</i> ∘ <i>h</i></span>. The chain rule states that to compute the derivative of <span class="texhtml"><i>f</i> ∘ <i>g</i> ∘ <i>h</i></span>, it is sufficient to compute the derivative of <i><span class="texhtml mvar" style="font-style:italic;">f</span></i> and the derivative of <span class="texhtml"><i>g</i> ∘ <i>h</i></span>. The derivative of <span class="texhtml mvar" style="font-style:italic;">f</span> can be calculated directly, and the derivative of <span class="texhtml"><i>g</i> ∘ <i>h</i></span> can be calculated by applying the chain rule again.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2023)">citation needed</span></a></i>]</sup> </p><p>For concreteness, consider the function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=e^{\sin(x^{2})}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=e^{\sin(x^{2})}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97833e8b3754a0687e7aa2c5d19e8c458b1d7537" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.287ex; height:3.343ex;" alt="{\displaystyle y=e^{\sin(x^{2})}.}" /></span> This can be decomposed as the composite of three functions: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}y&=f(u)=e^{u},\\u&=g(v)=\sin v,\\v&=h(x)=x^{2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>v</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y&=f(u)=e^{u},\\u&=g(v)=\sin v,\\v&=h(x)=x^{2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f43f236df66b883211c831396c4bc022ccb062" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.348ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}y&=f(u)=e^{u},\\u&=g(v)=\sin v,\\v&=h(x)=x^{2}.\end{aligned}}}" /></span> So that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(g(h(x)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(g(h(x)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71908fdc47466c68e908ae51a119cf7258990ab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.745ex; height:2.843ex;" alt="{\displaystyle y=f(g(h(x)))}" /></span>. </p><p>Their derivatives are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {dy}{du}}&=f'(u)=e^{u},\\{\frac {du}{dv}}&=g'(v)=\cos v,\\{\frac {dv}{dx}}&=h'(x)=2x.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>v</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dy}{du}}&=f'(u)=e^{u},\\{\frac {du}{dv}}&=g'(v)=\cos v,\\{\frac {dv}{dx}}&=h'(x)=2x.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7364d5fbe74c415b0b60b9cd154b3cc0033b89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:20.343ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {dy}{du}}&=f'(u)=e^{u},\\{\frac {du}{dv}}&=g'(v)=\cos v,\\{\frac {dv}{dx}}&=h'(x)=2x.\end{aligned}}}" /></span> </p><p>The chain rule states that the derivative of their composite at the point <span class="texhtml"><i>x</i> = <i>a</i></span> is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=f'((g\circ h)(a))\cdot (g\circ h)'(a)\\&=f'((g\circ h)(a))\cdot g'(h(a))\cdot h'(a)\\&=(f'\circ g\circ h)(a)\cdot (g'\circ h)(a)\cdot h'(a).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=f'((g\circ h)(a))\cdot (g\circ h)'(a)\\&=f'((g\circ h)(a))\cdot g'(h(a))\cdot h'(a)\\&=(f'\circ g\circ h)(a)\cdot (g'\circ h)(a)\cdot h'(a).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f761436494d4651da92ec072bc8fa188404d77a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:50.457ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=f'((g\circ h)(a))\cdot (g\circ h)'(a)\\&=f'((g\circ h)(a))\cdot g'(h(a))\cdot h'(a)\\&=(f'\circ g\circ h)(a)\cdot (g'\circ h)(a)\cdot h'(a).\end{aligned}}}" /></span> </p><p>In <a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a>, this is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dx}}=\left.{\frac {dy}{du}}\right|_{u=g(h(a))}\cdot \left.{\frac {du}{dv}}\right|_{v=h(a)}\cdot \left.{\frac {dv}{dx}}\right|_{x=a},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋅</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋅</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=\left.{\frac {dy}{du}}\right|_{u=g(h(a))}\cdot \left.{\frac {du}{dv}}\right|_{v=h(a)}\cdot \left.{\frac {dv}{dx}}\right|_{x=a},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d29f7dfbbb2ec14ab9d4231d1978e208e71a8bdd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.911ex; height:6.343ex;" alt="{\displaystyle {\frac {dy}{dx}}=\left.{\frac {dy}{du}}\right|_{u=g(h(a))}\cdot \left.{\frac {du}{dv}}\right|_{v=h(a)}\cdot \left.{\frac {dv}{dx}}\right|_{x=a},}" /></span> or for short, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dv}}\cdot {\frac {dv}{dx}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dv}}\cdot {\frac {dv}{dx}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe662eb086c4fb4037a83210f9df0797d83f124c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.63ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dv}}\cdot {\frac {dv}{dx}}.}" /></span> The derivative function is therefore: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dx}}=e^{\sin(x^{2})}\cdot \cos(x^{2})\cdot 2x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>2</mn> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=e^{\sin(x^{2})}\cdot \cos(x^{2})\cdot 2x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce43f2e5a0a2286272284ceb8392218f4acd7aed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.668ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dx}}=e^{\sin(x^{2})}\cdot \cos(x^{2})\cdot 2x.}" /></span> </p><p>Another way of computing this derivative is to view the composite function <span class="texhtml"><i>f</i> ∘ <i>g</i> ∘ <i>h</i></span> as the composite of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> and <i>h</i>. Applying the chain rule in this manner would yield: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=(f\circ g)'(h(a))\cdot h'(a)\\&=f'(g(h(a)))\cdot g'(h(a))\cdot h'(a).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>∘</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=(f\circ g)'(h(a))\cdot h'(a)\\&=f'(g(h(a)))\cdot g'(h(a))\cdot h'(a).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4a777f9abc043667511df6d992e8374ea7530d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.682ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=(f\circ g)'(h(a))\cdot h'(a)\\&=f'(g(h(a)))\cdot g'(h(a))\cdot h'(a).\end{aligned}}}" /></span> </p><p>This is the same as what was computed above. This should be expected because <span class="texhtml">(<i>f</i> ∘ <i>g</i>) ∘ <i>h</i> = <i>f</i> ∘ (<i>g</i> ∘ <i>h</i>)</span>. </p><p>Sometimes, it is necessary to differentiate an arbitrarily long composition of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\circ f_{2}\circ \cdots \circ f_{n-1}\circ f_{n}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}\circ f_{2}\circ \cdots \circ f_{n-1}\circ f_{n}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad17774b1e18d7afab5db60bcf4136c48523918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:22.705ex; height:2.509ex;" alt="{\displaystyle f_{1}\circ f_{2}\circ \cdots \circ f_{n-1}\circ f_{n}\!}" /></span>. In this case, define <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{a\,.\,.\,b}=f_{a}\circ f_{a+1}\circ \cdots \circ f_{b-1}\circ f_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{a\,.\,.\,b}=f_{a}\circ f_{a+1}\circ \cdots \circ f_{b-1}\circ f_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72de180ec13c68b5b19714060b82ce1c4b9e759b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.46ex; height:2.509ex;" alt="{\displaystyle f_{a\,.\,.\,b}=f_{a}\circ f_{a+1}\circ \cdots \circ f_{b-1}\circ f_{b}}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{a\,.\,.\,a}=f_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{a\,.\,.\,a}=f_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b8d23262548189fe7f5978501390f3b46efe30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.526ex; height:2.509ex;" alt="{\displaystyle f_{a\,.\,.\,a}=f_{a}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{a\,.\,.\,b}(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{a\,.\,.\,b}(x)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3e961622981a4341a3f49ecbc95d7c97fa0ffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.59ex; height:2.843ex;" alt="{\displaystyle f_{a\,.\,.\,b}(x)=x}" /></span> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec536027666ac731c00e6db3a3b0b10d7ebbe031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b<a}" /></span>. Then the chain rule takes the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}Df_{1\,.\,.\,n}&=(Df_{1}\circ f_{2\,.\,.\,n})(Df_{2}\circ f_{3\,.\,.\,n})\cdots (Df_{n-1}\circ f_{n\,.\,.\,n})Df_{n}\\&=\prod _{k=1}^{n}\left[Df_{k}\circ f_{(k+1)\,.\,.\,n}\right]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>D</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>D</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>D</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>D</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>D</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <mi>D</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Df_{1\,.\,.\,n}&=(Df_{1}\circ f_{2\,.\,.\,n})(Df_{2}\circ f_{3\,.\,.\,n})\cdots (Df_{n-1}\circ f_{n\,.\,.\,n})Df_{n}\\&=\prod _{k=1}^{n}\left[Df_{k}\circ f_{(k+1)\,.\,.\,n}\right]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0158351dac8c146e0f3fe16806c991851824226" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:61.371ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}Df_{1\,.\,.\,n}&=(Df_{1}\circ f_{2\,.\,.\,n})(Df_{2}\circ f_{3\,.\,.\,n})\cdots (Df_{n-1}\circ f_{n\,.\,.\,n})Df_{n}\\&=\prod _{k=1}^{n}\left[Df_{k}\circ f_{(k+1)\,.\,.\,n}\right]\end{aligned}}}" /></span> or, in the Lagrange notation, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f_{1\,.\,.\,n}'(x)&=f_{1}'\left(f_{2\,.\,.\,n}(x)\right)\;f_{2}'\left(f_{3\,.\,.\,n}(x)\right)\cdots f_{n-1}'\left(f_{n\,.\,.\,n}(x)\right)\;f_{n}'(x)\\[1ex]&=\prod _{k=1}^{n}f_{k}'\left(f_{(k+1\,.\,.\,n)}(x)\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace"></mspace> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace"></mspace> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{1\,.\,.\,n}'(x)&=f_{1}'\left(f_{2\,.\,.\,n}(x)\right)\;f_{2}'\left(f_{3\,.\,.\,n}(x)\right)\cdots f_{n-1}'\left(f_{n\,.\,.\,n}(x)\right)\;f_{n}'(x)\\[1ex]&=\prod _{k=1}^{n}f_{k}'\left(f_{(k+1\,.\,.\,n)}(x)\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f97176b768ccee32494a7e77e30e8820d90740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:64.086ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}f_{1\,.\,.\,n}'(x)&=f_{1}'\left(f_{2\,.\,.\,n}(x)\right)\;f_{2}'\left(f_{3\,.\,.\,n}(x)\right)\cdots f_{n-1}'\left(f_{n\,.\,.\,n}(x)\right)\;f_{n}'(x)\\[1ex]&=\prod _{k=1}^{n}f_{k}'\left(f_{(k+1\,.\,.\,n)}(x)\right)\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Quotient_rule">Quotient rule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=6" title="Edit section: Quotient rule"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Quotient_rule" title="Quotient rule">Quotient rule</a></div> <p>The chain rule can be used to derive some well-known differentiation rules. For example, the quotient rule is a consequence of the chain rule and the <a href="/wiki/Product_rule" title="Product rule">product rule</a>. To see this, write the function <span class="texhtml"><i>f</i>(<i>x</i>)/<i>g</i>(<i>x</i>)</span> as the product <span class="texhtml"><i>f</i>(<i>x</i>) · 1/<i>g</i>(<i>x</i>)</span>. First apply the product rule: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)&={\frac {d}{dx}}\left(f(x)\cdot {\frac {1}{g(x)}}\right)\\&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)&={\frac {d}{dx}}\left(f(x)\cdot {\frac {1}{g(x)}}\right)\\&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716b23f28ab8006d33c15229ef6e7f013dc8fa6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:50.46ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)&={\frac {d}{dx}}\left(f(x)\cdot {\frac {1}{g(x)}}\right)\\&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right).\end{aligned}}}" /></span> </p><p>To compute the derivative of <span class="texhtml">1/<i>g</i>(<i>x</i>)</span>, notice that it is the composite of <span class="texhtml mvar" style="font-style:italic;">g</span> with the reciprocal function, that is, the function that sends <span class="texhtml mvar" style="font-style:italic;">x</span> to <span class="texhtml">1/<i>x</i></span>. The derivative of the reciprocal function is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1/x^{2}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1/x^{2}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f53e97fca27e254f31b78e2add9d21ba903a855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:6.517ex; height:3.176ex;" alt="{\displaystyle -1/x^{2}\!}" /></span>. By applying the chain rule, the last expression becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot \left(-{\frac {1}{g(x)^{2}}}\cdot g'(x)\right)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot \left(-{\frac {1}{g(x)^{2}}}\cdot g'(x)\right)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc8fcb12f6b86d7094b4286b92627808a16aa8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:65.028ex; height:6.509ex;" alt="{\displaystyle f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot \left(-{\frac {1}{g(x)^{2}}}\cdot g'(x)\right)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}},}" /></span> which is the usual formula for the quotient rule. </p> <div class="mw-heading mw-heading3"><h3 id="Derivatives_of_inverse_functions">Derivatives of inverse functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=7" title="Edit section: Derivatives of inverse functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></div> <p>Suppose that <span class="texhtml"><i>y</i> = <i>g</i>(<i>x</i>)</span> has an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>. Call its inverse function <span class="texhtml mvar" style="font-style:italic;">f</span> so that we have <span class="texhtml"><i>x</i> = <i>f</i>(<i>y</i>)</span>. There is a formula for the derivative of <span class="texhtml mvar" style="font-style:italic;">f</span> in terms of the derivative of <span class="texhtml mvar" style="font-style:italic;">g</span>. To see this, note that <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> satisfy the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(x))=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(x))=x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d074a49edeb387d1b028abdde5dc6462351e52" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.418ex; height:2.843ex;" alt="{\displaystyle f(g(x))=x.}" /></span> </p><p>And because the functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d3aa048bd1973b529fafede33c59f18b4058d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:2.843ex;" alt="{\displaystyle f(g(x))}" /></span> and <span class="texhtml mvar" style="font-style:italic;">x</span> are equal, their derivatives must be equal. The derivative of <span class="texhtml mvar" style="font-style:italic;">x</span> is the constant function with value 1, and the derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d3aa048bd1973b529fafede33c59f18b4058d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:2.843ex;" alt="{\displaystyle f(g(x))}" /></span> is determined by the chain rule. Therefore, we have that: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(g(x))g'(x)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(g(x))g'(x)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/758a6cfad399b929aee90f2624b1e81a24b0d9c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.919ex; height:3.009ex;" alt="{\displaystyle f'(g(x))g'(x)=1.}" /></span> </p><p>To express <span class="texhtml mvar" style="font-style:italic;">f'</span> as a function of an independent variable <span class="texhtml mvar" style="font-style:italic;">y</span>, we substitute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa9215c6afa4892692ba05ae4c44f23600ea79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.243ex; height:2.843ex;" alt="{\displaystyle f(y)}" /></span> for <span class="texhtml mvar" style="font-style:italic;">x</span> wherever it appears. Then we can solve for <span class="texhtml mvar" style="font-style:italic;">f'</span>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f'(g(f(y)))g'(f(y))&=1\\f'(y)g'(f(y))&=1\\f'(y)={\frac {1}{g'(f(y))}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f'(g(f(y)))g'(f(y))&=1\\f'(y)g'(f(y))&=1\\f'(y)={\frac {1}{g'(f(y))}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7aeaf7b2d7e0779851ec881d454d992bd012c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:23.851ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}f'(g(f(y)))g'(f(y))&=1\\f'(y)g'(f(y))&=1\\f'(y)={\frac {1}{g'(f(y))}}.\end{aligned}}}" /></span> </p><p>For example, consider the function <span class="texhtml"><i>g</i>(<i>x</i>) = <i>e</i><sup><i>x</i></sup></span>. It has an inverse <span class="texhtml"><i>f</i>(<i>y</i>) = ln <i>y</i></span>. Because <span class="texhtml"><i>g</i>′(<i>x</i>) = <i>e</i><sup><i>x</i></sup></span>, the above formula says that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dy}}\ln y={\frac {1}{e^{\ln y}}}={\frac {1}{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dy}}\ln y={\frac {1}{e^{\ln y}}}={\frac {1}{y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c4f7d978c19f4a679c339d52d636849c5f6c20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.646ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dy}}\ln y={\frac {1}{e^{\ln y}}}={\frac {1}{y}}.}" /></span> </p><p>This formula is true whenever <span class="texhtml mvar" style="font-style:italic;">g</span> is differentiable and its inverse <span class="texhtml mvar" style="font-style:italic;">f</span> is also differentiable. This formula can fail when one of these conditions is not true. For example, consider <span class="texhtml"><i>g</i>(<i>x</i>) = <i>x</i><sup>3</sup></span>. Its inverse is <span class="texhtml"><i>f</i>(<i>y</i>) = <i>y</i><sup>1/3</sup></span>, which is not differentiable at zero. If we attempt to use the above formula to compute the derivative of <span class="texhtml mvar" style="font-style:italic;">f</span> at zero, then we must evaluate <span class="texhtml">1/<i>g</i>′(<i>f</i>(0))</span>. Since <span class="texhtml"><i>f</i>(0) = 0</span> and <span class="texhtml"><i>g</i>′(0) = 0</span>, we must evaluate 1/0, which is undefined. Therefore, the formula fails in this case. This is not surprising because <span class="texhtml mvar" style="font-style:italic;">f</span> is not differentiable at zero. </p> <div class="mw-heading mw-heading3"><h3 id="Back_propagation">Back propagation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=8" title="Edit section: Back propagation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The chain rule forms the basis of the <a href="/wiki/Back_propagation" class="mw-redirect" title="Back propagation">back propagation</a> algorithm, which is used in <a href="/wiki/Gradient_descent" title="Gradient descent">gradient descent</a> of <a href="/wiki/Neural_network_(machine_learning)" title="Neural network (machine learning)">neural networks</a> in <a href="/wiki/Deep_learning" title="Deep learning">deep learning</a> (<a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Higher_derivatives">Higher derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=9" title="Edit section: Higher derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a> generalizes the chain rule to higher derivatives. Assuming that <span class="texhtml"><i>y</i> = <i>f</i>(<i>u</i>)</span> and <span class="texhtml"><i>u</i> = <i>g</i>(<i>x</i>)</span>, then the first few derivatives are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e3f29cf2b81e2d94439ea0a21b912f518ef3ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:87.025ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Proofs">Proofs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=10" title="Edit section: Proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="First_proof">First proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=11" title="Edit section: First proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One proof of the chain rule begins by defining the derivative of the composite function <span class="texhtml"><i>f</i> ∘ <i>g</i></span>, where we take the <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a> of the <a href="/wiki/Difference_quotient" title="Difference quotient">difference quotient</a> for <span class="texhtml"><i>f</i> ∘ <i>g</i></span> as <span class="texhtml mvar" style="font-style:italic;">x</span> approaches <span class="texhtml mvar" style="font-style:italic;">a</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ g)'(a)=\lim _{x\to a}{\frac {f(g(x))-f(g(a))}{x-a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\circ g)'(a)=\lim _{x\to a}{\frac {f(g(x))-f(g(a))}{x-a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c73c53631829c20566f587ca560394048d69f99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.97ex; height:5.843ex;" alt="{\displaystyle (f\circ g)'(a)=\lim _{x\to a}{\frac {f(g(x))-f(g(a))}{x-a}}.}" /></span> </p><p>Assume for the moment that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadd91c96783e237ee7520e413c7fce51c333563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:4.034ex; height:2.843ex;" alt="{\displaystyle g(x)\!}" /></span> does not equal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7205153da0f7522baf23e2a109ba6630e8104c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.155ex; height:2.843ex;" alt="{\displaystyle g(a)}" /></span> for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> near <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span>. Then the previous expression is equal to the product of two factors: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to a}{\frac {f(g(x))-f(g(a))}{g(x)-g(a)}}\cdot {\frac {g(x)-g(a)}{x-a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a}{\frac {f(g(x))-f(g(a))}{g(x)-g(a)}}\cdot {\frac {g(x)-g(a)}{x-a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0db90d4c74d36faad7007f20c561fa9d9244643" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.515ex; height:6.509ex;" alt="{\displaystyle \lim _{x\to a}{\frac {f(g(x))-f(g(a))}{g(x)-g(a)}}\cdot {\frac {g(x)-g(a)}{x-a}}.}" /></span> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /></span> oscillates near <span class="texhtml mvar" style="font-style:italic;">a</span>, then it might happen that no matter how close one gets to <span class="texhtml mvar" style="font-style:italic;">a</span>, there is always an even closer <span class="texhtml mvar" style="font-style:italic;">x</span> such that <span class="texhtml"><i>g</i>(<i>x</i>) = <i>g</i>(<i>a</i>)</span>. For example, this happens near <span class="texhtml"><i>a</i> = 0</span> for the <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> <span class="texhtml mvar" style="font-style:italic;">g</span> defined by <span class="texhtml"><i>g</i>(<i>x</i>) = 0</span> for <span class="texhtml"><i>x</i> = 0</span> and <span class="texhtml"><i>g</i>(<i>x</i>) = <i>x</i><sup>2</sup> sin(1/<i>x</i>)</span> otherwise. Whenever this happens, the above expression is undefined because it involves <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>. To work around this, introduce a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" /></span> as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(y)={\begin{cases}\displaystyle {\frac {f(y)-f(g(a))}{y-g(a)}},&y\neq g(a),\\f'(g(a)),&y=g(a).\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>y</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mtd> <mtd> <mi>y</mi> <mo>≠</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(y)={\begin{cases}\displaystyle {\frac {f(y)-f(g(a))}{y-g(a)}},&y\neq g(a),\\f'(g(a)),&y=g(a).\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d67de682aed22022801312a7116e594ad0a5124" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.907ex; height:9.176ex;" alt="{\displaystyle Q(y)={\begin{cases}\displaystyle {\frac {f(y)-f(g(a))}{y-g(a)}},&y\neq g(a),\\f'(g(a)),&y=g(a).\end{cases}}}" /></span> We will show that the difference quotient for <span class="texhtml"><i>f</i> ∘ <i>g</i></span> is always equal to: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(g(x))\cdot {\frac {g(x)-g(a)}{x-a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(g(x))\cdot {\frac {g(x)-g(a)}{x-a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27673843502ae87de9751bbf382d3be81be16bf3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.315ex; height:5.843ex;" alt="{\displaystyle Q(g(x))\cdot {\frac {g(x)-g(a)}{x-a}}.}" /></span> </p><p>Whenever <span class="texhtml"><i>g</i>(<i>x</i>)</span> is not equal to <span class="texhtml"><i>g</i>(<i>a</i>)</span>, this is clear because the factors of <span class="texhtml"><i>g</i>(<i>x</i>) − <i>g</i>(<i>a</i>)</span> cancel. When <span class="texhtml"><i>g</i>(<i>x</i>)</span> equals <span class="texhtml"><i>g</i>(<i>a</i>)</span>, then the difference quotient for <span class="texhtml"><i>f</i> ∘ <i>g</i></span> is zero because <span class="texhtml"><i>f</i>(<i>g</i>(<i>x</i>))</span> equals <span class="texhtml"><i>f</i>(<i>g</i>(<i>a</i>))</span>, and the above product is zero because it equals <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))</span> times zero. So the above product is always equal to the difference quotient, and to show that the derivative of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> at <span class="texhtml"><i>a</i></span> exists and to determine its value, we need only show that the limit as <span class="texhtml"><i>x</i></span> goes to <span class="texhtml"><i>a</i></span> of the above product exists and determine its value. </p><p>To do this, recall that the limit of a product exists if the limits of its factors exist. When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are <span class="texhtml"><i>Q</i>(<i>g</i>(<i>x</i>))</span> and <span class="texhtml">(<i>g</i>(<i>x</i>) − <i>g</i>(<i>a</i>)) / (<i>x</i> − <i>a</i>)</span>. The latter is the difference quotient for <span class="texhtml mvar" style="font-style:italic;">g</span> at <span class="texhtml mvar" style="font-style:italic;">a</span>, and because <span class="texhtml mvar" style="font-style:italic;">g</span> is differentiable at <span class="texhtml mvar" style="font-style:italic;">a</span> by assumption, its limit as <span class="texhtml mvar" style="font-style:italic;">x</span> tends to <span class="texhtml mvar" style="font-style:italic;">a</span> exists and equals <span class="texhtml"><i>g</i>′(<i>a</i>)</span>. </p><p>As for <span class="texhtml"><i>Q</i>(<i>g</i>(<i>x</i>))</span>, notice that <span class="texhtml"><i>Q</i></span> is defined wherever <i><span class="texhtml mvar" style="font-style:italic;">f</span></i> is. Furthermore, <i><span class="texhtml mvar" style="font-style:italic;">f</span></i> is differentiable at <span class="texhtml"><i>g</i>(<i>a</i>)</span> by assumption, so <span class="texhtml"><i>Q</i></span> is continuous at <span class="texhtml"><i>g</i>(<i>a</i>)</span>, by definition of the derivative. The function <span class="texhtml mvar" style="font-style:italic;">g</span> is continuous at <span class="texhtml mvar" style="font-style:italic;">a</span> because it is differentiable at <span class="texhtml mvar" style="font-style:italic;">a</span>, and therefore <span class="texhtml"><i>Q</i> ∘ <i>g</i></span> is continuous at <span class="texhtml mvar" style="font-style:italic;">a</span>. So its limit as <i><span class="texhtml mvar" style="font-style:italic;">x</span></i> goes to <i><span class="texhtml mvar" style="font-style:italic;">a</span></i> exists and equals <span class="texhtml"><i>Q</i>(<i>g</i>(<i>a</i>))</span>, which is <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))</span>. </p><p>This shows that the limits of both factors exist and that they equal <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))</span> and <span class="texhtml"><i>g</i>′(<i>a</i>)</span>, respectively. Therefore, the derivative of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> at <i>a</i> exists and equals <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))</span><span class="texhtml"><i>g</i>′(<i>a</i>)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Second_proof">Second proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=12" title="Edit section: Second proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative. This proof has the advantage that it generalizes to several variables. It relies on the following equivalent definition of differentiability at a point: A function <i>g</i> is differentiable at <i>a</i> if there exists a real number <i>g</i>′(<i>a</i>) and a function <i>ε</i>(<i>h</i>) that tends to zero as <i>h</i> tends to zero, and furthermore <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(a+h)-g(a)=g'(a)h+\varepsilon (h)h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(a+h)-g(a)=g'(a)h+\varepsilon (h)h.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8bedc0b12b3ab500a4e21f269b3efa1f5112b89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.667ex; height:3.009ex;" alt="{\displaystyle g(a+h)-g(a)=g'(a)h+\varepsilon (h)h.}" /></span> Here the left-hand side represents the true difference between the value of <i>g</i> at <i>a</i> and at <span class="texhtml"><i>a</i> + <i>h</i></span>, whereas the right-hand side represents the approximation determined by the derivative plus an error term. </p><p>In the situation of the chain rule, such a function <i>ε</i> exists because <i>g</i> is assumed to be differentiable at <i>a</i>. Again by assumption, a similar function also exists for <i>f</i> at <i>g</i>(<i>a</i>). Calling this function <i>η</i>, we have <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(a)+k)-f(g(a))=f'(g(a))k+\eta (k)k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>k</mi> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(a)+k)-f(g(a))=f'(g(a))k+\eta (k)k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04426654ce142285b833315752b500507b2ea0c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.546ex; height:3.009ex;" alt="{\displaystyle f(g(a)+k)-f(g(a))=f'(g(a))k+\eta (k)k.}" /></span> The above definition imposes no constraints on <i>η</i>(0), even though it is assumed that <i>η</i>(<i>k</i>) tends to zero as <i>k</i> tends to zero. If we set <span class="texhtml"><i>η</i>(0) = 0</span>, then <i>η</i> is continuous at 0. </p><p>Proving the theorem requires studying the difference <span class="texhtml"><i>f</i>(<i>g</i>(<i>a</i> + <i>h</i>)) − <i>f</i>(<i>g</i>(<i>a</i>))</span> as <i>h</i> tends to zero. The first step is to substitute for <span class="texhtml"><i>g</i>(<i>a</i> + <i>h</i>)</span> using the definition of differentiability of <i>g</i> at <i>a</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(a+h))-f(g(a))=f(g(a)+g'(a)h+\varepsilon (h)h)-f(g(a)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(a+h))-f(g(a))=f(g(a)+g'(a)h+\varepsilon (h)h)-f(g(a)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04addd0dff3b58fdefa7cb0c5dfc337b9c413de8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.01ex; height:3.009ex;" alt="{\displaystyle f(g(a+h))-f(g(a))=f(g(a)+g'(a)h+\varepsilon (h)h)-f(g(a)).}" /></span> The next step is to use the definition of differentiability of <i>f</i> at <i>g</i>(<i>a</i>). This requires a term of the form <span class="texhtml"><i>f</i>(<i>g</i>(<i>a</i>) + <i>k</i>)</span> for some <i>k</i>. In the above equation, the correct <i>k</i> varies with <i>h</i>. Set <span class="texhtml"><i>k</i><sub><i>h</i></sub> = <i>g</i>′(<i>a</i>) <i>h</i> + <i>ε</i>(<i>h</i>) <i>h</i></span> and the right hand side becomes <span class="texhtml"><i>f</i>(<i>g</i>(<i>a</i>) + <i>k</i><sub><i>h</i></sub>) − <i>f</i>(<i>g</i>(<i>a</i>))</span>. Applying the definition of the derivative gives: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(a)+k_{h})-f(g(a))=f'(g(a))k_{h}+\eta (k_{h})k_{h}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(a)+k_{h})-f(g(a))=f'(g(a))k_{h}+\eta (k_{h})k_{h}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcbfb77bbcc2ef923ca65b07931be806b70e92e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.262ex; height:3.009ex;" alt="{\displaystyle f(g(a)+k_{h})-f(g(a))=f'(g(a))k_{h}+\eta (k_{h})k_{h}.}" /></span> To study the behavior of this expression as <i>h</i> tends to zero, expand <i>k</i><sub><i>h</i></sub>. After regrouping the terms, the right-hand side becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(g(a))g'(a)h+[f'(g(a))\varepsilon (h)+\eta (k_{h})g'(a)+\eta (k_{h})\varepsilon (h)]h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo>+</mo> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(g(a))g'(a)h+[f'(g(a))\varepsilon (h)+\eta (k_{h})g'(a)+\eta (k_{h})\varepsilon (h)]h.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5298f1d5c0274ae4d053106e9649c4f25503f9c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.965ex; height:3.009ex;" alt="{\displaystyle f'(g(a))g'(a)h+[f'(g(a))\varepsilon (h)+\eta (k_{h})g'(a)+\eta (k_{h})\varepsilon (h)]h.}" /></span> Because <i>ε</i>(<i>h</i>) and <i>η</i>(<i>k</i><sub><i>h</i></sub>) tend to zero as <i>h</i> tends to zero, the first two bracketed terms tend to zero as <i>h</i> tends to zero. Applying the same theorem on products of limits as in the first proof, the third bracketed term also tends zero. Because the above expression is equal to the difference <span class="texhtml"><i>f</i>(<i>g</i>(<i>a</i> + <i>h</i>)) − <i>f</i>(<i>g</i>(<i>a</i>))</span>, by the definition of the derivative <span class="texhtml"><i>f</i> ∘ <i>g</i></span> is differentiable at <i>a</i> and its derivative is <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>)) <i>g</i>′(<i>a</i>).</span> </p><p>The role of <i>Q</i> in the first proof is played by <i>η</i> in this proof. They are related by the equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(y)=f'(g(a))+\eta (y-g(a)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(y)=f'(g(a))+\eta (y-g(a)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d631170ae01fab438bf07c67ba792b5b80b671c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.488ex; height:3.009ex;" alt="{\displaystyle Q(y)=f'(g(a))+\eta (y-g(a)).}" /></span> The need to define <i>Q</i> at <i>g</i>(<i>a</i>) is analogous to the need to define <i>η</i> at zero. </p> <div class="mw-heading mw-heading3"><h3 id="Third_proof">Third proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=13" title="Edit section: Third proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Constantin Carathéodory</a>'s alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Under this definition, a function <span class="texhtml mvar" style="font-style:italic;">f</span> is differentiable at a point <span class="texhtml mvar" style="font-style:italic;">a</span> if and only if there is a function <span class="texhtml mvar" style="font-style:italic;">q</span>, continuous at <span class="texhtml mvar" style="font-style:italic;">a</span> and such that <span class="texhtml"><i>f</i>(<i>x</i>) − <i>f</i>(<i>a</i>) = <i>q</i>(<i>x</i>)(<i>x</i> − <i>a</i>)</span>. There is at most one such function, and if <span class="texhtml mvar" style="font-style:italic;">f</span> is differentiable at <span class="texhtml mvar" style="font-style:italic;">a</span> then <span class="texhtml"><i>f</i> ′(<i>a</i>) = <i>q</i>(<i>a</i>)</span>. </p><p>Given the assumptions of the chain rule and the fact that differentiable functions and compositions of continuous functions are continuous, we have that there exist functions <span class="texhtml mvar" style="font-style:italic;">q</span>, continuous at <span class="texhtml"><i>g</i>(<i>a</i>)</span>, and <span class="texhtml mvar" style="font-style:italic;">r</span>, continuous at <span class="texhtml mvar" style="font-style:italic;">a</span>, and such that, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(x))-f(g(a))=q(g(x))(g(x)-g(a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(x))-f(g(a))=q(g(x))(g(x)-g(a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfe8c9dac59735d4f0bdc94e1e89b4150ac25e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.718ex; height:2.843ex;" alt="{\displaystyle f(g(x))-f(g(a))=q(g(x))(g(x)-g(a))}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)-g(a)=r(x)(x-a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)-g(a)=r(x)(x-a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbe2848af662d61b739dd1dff93647a09c59b36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.393ex; height:2.843ex;" alt="{\displaystyle g(x)-g(a)=r(x)(x-a).}" /></span> Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g(x))-f(g(a))=q(g(x))r(x)(x-a),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g(x))-f(g(a))=q(g(x))r(x)(x-a),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bfbfaf8dc2f9a225cd748f1d46547cc1f738d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.702ex; height:2.843ex;" alt="{\displaystyle f(g(x))-f(g(a))=q(g(x))r(x)(x-a),}" /></span> but the function given by <span class="texhtml"><i>h</i>(<i>x</i>) = <i>q</i>(<i>g</i>(<i>x</i>))<i>r</i>(<i>x</i>)</span> is continuous at <span class="texhtml mvar" style="font-style:italic;">a</span>, and we get, for this <span class="texhtml mvar" style="font-style:italic;">a</span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(g(a)))'=q(g(a))r(a)=f'(g(a))g'(a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f(g(a)))'=q(g(a))r(a)=f'(g(a))g'(a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e6fc00c01bcdb640b0c24c63da7e161525a7ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.514ex; height:3.009ex;" alt="{\displaystyle (f(g(a)))'=q(g(a))r(a)=f'(g(a))g'(a).}" /></span> A similar approach works for continuously differentiable (vector-)functions of many variables. This method of factoring also allows a unified approach to stronger forms of differentiability, when the derivative is required to be <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuous</a>, <a href="/wiki/H%C3%B6lder_condition" title="Hölder condition">Hölder continuous</a>, etc. Differentiation itself can be viewed as the <a href="/wiki/Polynomial_remainder_theorem" title="Polynomial remainder theorem">polynomial remainder theorem</a> (the little <a href="/wiki/%C3%89tienne_B%C3%A9zout" title="Étienne Bézout">Bézout</a> theorem, or factor theorem), generalized to an appropriate class of functions.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2016)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Proof_via_infinitesimals">Proof via infinitesimals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=14" title="Edit section: Proof via infinitesimals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Non-standard_calculus" class="mw-redirect" title="Non-standard calculus">Non-standard calculus</a></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=g(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a8c69e0007ccd029aef87a5520e51388552e93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.193ex; height:2.843ex;" alt="{\displaystyle x=g(t)}" /></span> then choosing infinitesimal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t\not =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t\not =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db00e6d8d2c4f6f419281b16d27077941d9307b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.036ex; height:2.676ex;" alt="{\displaystyle \Delta t\not =0}" /></span> we compute the corresponding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x=g(t+\Delta t)-g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=g(t+\Delta t)-g(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588d35bde93b4099f93a665287ac1b7c4c93c93e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.35ex; height:2.843ex;" alt="{\displaystyle \Delta x=g(t+\Delta t)-g(t)}" /></span> and then the corresponding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y=f(x+\Delta x)-f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y=f(x+\Delta x)-f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce43d41f207f5cdc2737e774b438b6cee1aa7401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.971ex; height:2.843ex;" alt="{\displaystyle \Delta y=f(x+\Delta x)-f(x)}" /></span>, so that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta y}{\Delta t}}={\frac {\Delta y}{\Delta x}}{\frac {\Delta x}{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta y}{\Delta t}}={\frac {\Delta y}{\Delta x}}{\frac {\Delta x}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bda240671122c175309ecb2f18f095afc7838ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.229ex; height:5.676ex;" alt="{\displaystyle {\frac {\Delta y}{\Delta t}}={\frac {\Delta y}{\Delta x}}{\frac {\Delta x}{\Delta t}}}" /></span> and applying the <a href="/wiki/Standard_part" class="mw-redirect" title="Standard part">standard part</a> we obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}{\frac {dx}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}{\frac {dx}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f092ea1b2f4dab2125e06931a9ccf948b6b0526" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.069ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}{\frac {dx}{dt}}}" /></span> which is the chain rule. </p> <div class="mw-heading mw-heading2"><h2 id="Multivariable_case">Multivariable case</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=15" title="Edit section: Multivariable case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The full generalization of the chain rule to <a href="/wiki/Multi-variable_function" class="mw-redirect" title="Multi-variable function">multi-variable functions</a> (such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fa8a698b05433f8431190a957e0f413cda8e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}" /></span>) is rather technical. However, it is simpler to write in the case of functions of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g_{1}(x),\dots ,g_{k}(x)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g_{1}(x),\dots ,g_{k}(x)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122f004a316cd58b1ef7114d16885e495036f01d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.552ex; height:2.843ex;" alt="{\displaystyle f(g_{1}(x),\dots ,g_{k}(x)),}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c3f7ab7e2778b86252a09320aaa37ca541b462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.275ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afad54799de7468671e4e0c74ba85780279130cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.816ex; height:2.509ex;" alt="{\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} }" /></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,\dots ,k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,\dots ,k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d622e4c4b929ebe3c8481bffb011994347c1045c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.296ex; height:2.509ex;" alt="{\displaystyle i=1,2,\dots ,k.}" /></span> </p><p>As this case occurs often in the study of functions of a single variable, it is worth describing it separately. </p> <div class="mw-heading mw-heading3"><h3 id="Case_of_scalar-valued_functions_with_multiple_inputs">Case of scalar-valued functions with multiple inputs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=16" title="Edit section: Case of scalar-valued functions with multiple inputs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c3f7ab7e2778b86252a09320aaa37ca541b462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.275ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} }" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afad54799de7468671e4e0c74ba85780279130cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.816ex; height:2.509ex;" alt="{\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} }" /></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,\dots ,k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,\dots ,k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d622e4c4b929ebe3c8481bffb011994347c1045c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.296ex; height:2.509ex;" alt="{\displaystyle i=1,2,\dots ,k.}" /></span> To write the chain rule for the composition of functions <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto f(g_{1}(x),\dots ,g_{k}(x)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto f(g_{1}(x),\dots ,g_{k}(x)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d73916ccfb5b38a1cc69591f9e87657a3bb452" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.496ex; height:2.843ex;" alt="{\displaystyle x\mapsto f(g_{1}(x),\dots ,g_{k}(x)),}" /></span> one needs the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> of <span class="texhtml mvar" style="font-style:italic;">f</span> with respect to its <span class="texhtml mvar" style="font-style:italic;">k</span> arguments. The usual notations for partial derivatives involve names for the arguments of the function. As these arguments are not named in the above formula, it is simpler and clearer to use <a href="/wiki/Notation_for_differentiation#D-notation" title="Notation for differentiation"><i>D</i>-Notation</a>, and to denote by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6595a62c7d45801da5db9fac4c9588afee8051fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.003ex; height:2.509ex;" alt="{\displaystyle D_{i}f}" /></span> the partial derivative of <span class="texhtml mvar" style="font-style:italic;">f</span> with respect to its <span class="texhtml mvar" style="font-style:italic;">i</span>th argument, and by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}f(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}f(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/718696356eeea66358e1d2c99a77e3066fac3965" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.9ex; height:2.843ex;" alt="{\displaystyle D_{i}f(z)}" /></span> the value of this derivative at <span class="texhtml mvar" style="font-style:italic;">z</span>. </p><p>With this notation, the chain rule is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}f(g_{1}(x),\dots ,g_{k}(x))=\sum _{i=1}^{k}\left({\frac {d}{dx}}{g_{i}}(x)\right)D_{i}f(g_{1}(x),\dots ,g_{k}(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}f(g_{1}(x),\dots ,g_{k}(x))=\sum _{i=1}^{k}\left({\frac {d}{dx}}{g_{i}}(x)\right)D_{i}f(g_{1}(x),\dots ,g_{k}(x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e526246c1be1de64048bc7ce0436a7c96b6694e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.641ex; height:7.343ex;" alt="{\displaystyle {\frac {d}{dx}}f(g_{1}(x),\dots ,g_{k}(x))=\sum _{i=1}^{k}\left({\frac {d}{dx}}{g_{i}}(x)\right)D_{i}f(g_{1}(x),\dots ,g_{k}(x)).}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Example:_arithmetic_operations">Example: arithmetic operations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=17" title="Edit section: Example: arithmetic operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the function <span class="texhtml mvar" style="font-style:italic;">f</span> is addition, that is, if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u,v)=u+v,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u,v)=u+v,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cacd2b5d9be103aa2cd1f66494cae3ca9d304e6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.622ex; height:2.843ex;" alt="{\displaystyle f(u,v)=u+v,}" /></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle D_{1}f={\frac {\partial f}{\partial u}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle D_{1}f={\frac {\partial f}{\partial u}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68e58865b5907226389d61dc1015d72a4ecbfc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.325ex; height:4.176ex;" alt="{\textstyle D_{1}f={\frac {\partial f}{\partial u}}=1}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle D_{2}f={\frac {\partial f}{\partial v}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle D_{2}f={\frac {\partial f}{\partial v}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b125b5431e20b9f07038a5e5758395832f1548b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.289ex; height:4.176ex;" alt="{\textstyle D_{2}f={\frac {\partial f}{\partial v}}=1}" /></span>. Thus, the chain rule gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}(g(x)+h(x))=\left({\frac {d}{dx}}g(x)\right)D_{1}f+\left({\frac {d}{dx}}h(x)\right)D_{2}f={\frac {d}{dx}}g(x)+{\frac {d}{dx}}h(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}(g(x)+h(x))=\left({\frac {d}{dx}}g(x)\right)D_{1}f+\left({\frac {d}{dx}}h(x)\right)D_{2}f={\frac {d}{dx}}g(x)+{\frac {d}{dx}}h(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a015672d5ad24a12e07c7605c5c70066e72f94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.412ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dx}}(g(x)+h(x))=\left({\frac {d}{dx}}g(x)\right)D_{1}f+\left({\frac {d}{dx}}h(x)\right)D_{2}f={\frac {d}{dx}}g(x)+{\frac {d}{dx}}h(x).}" /></span> </p><p>For multiplication <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u,v)=uv,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u,v)=uv,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec5e827d41dcaf34e995608ff3b047d54885e4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.782ex; height:2.843ex;" alt="{\displaystyle f(u,v)=uv,}" /></span> the partials are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1}f=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{1}f=v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca8ab20159e9f8e495e11148a05eb1cf9dc3dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.483ex; height:2.509ex;" alt="{\displaystyle D_{1}f=v}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}f=u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{2}f=u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e92d2bc5551fef045b6814f737142d25f8b180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.685ex; height:2.509ex;" alt="{\displaystyle D_{2}f=u}" /></span>. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}(g(x)h(x))=h(x){\frac {d}{dx}}g(x)+g(x){\frac {d}{dx}}h(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}(g(x)h(x))=h(x){\frac {d}{dx}}g(x)+g(x){\frac {d}{dx}}h(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94626c6fe13e7d919a2138b2a845070d9c288fc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.739ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}(g(x)h(x))=h(x){\frac {d}{dx}}g(x)+g(x){\frac {d}{dx}}h(x).}" /></span> </p><p>The case of exponentiation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u,v)=u^{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(u,v)=u^{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c76b7003a9b4b1cd49f14d07e4f3679d8faa9f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.037ex; height:2.843ex;" alt="{\displaystyle f(u,v)=u^{v}}" /></span> is slightly more complicated, as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1}f=vu^{v-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mi>v</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{1}f=vu^{v-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a20dd40bc27317f09df289fd43dd990f0cd46e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.59ex; height:3.009ex;" alt="{\displaystyle D_{1}f=vu^{v-1},}" /></span> and, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{v}=e^{v\ln u},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>ln</mi> <mo>⁡</mo> <mi>u</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{v}=e^{v\ln u},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cdfbb7bc672576aa0dc44856c23f26e0bf3bc99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.303ex; height:3.009ex;" alt="{\displaystyle u^{v}=e^{v\ln u},}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}f=u^{v}\ln u.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>f</mi> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mi>ln</mi> <mo>⁡</mo> <mi>u</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{2}f=u^{v}\ln u.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04142deb6c65d6913d3bc1179660a70ee9bec833" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.405ex; height:2.676ex;" alt="{\displaystyle D_{2}f=u^{v}\ln u.}" /></span> It follows that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\left(g(x)^{h(x)}\right)=h(x)g(x)^{h(x)-1}{\frac {d}{dx}}g(x)+g(x)^{h(x)}\ln g(x)\,{\frac {d}{dx}}h(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>ln</mi> <mo>⁡</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\left(g(x)^{h(x)}\right)=h(x)g(x)^{h(x)-1}{\frac {d}{dx}}g(x)+g(x)^{h(x)}\ln g(x)\,{\frac {d}{dx}}h(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a90b2e29068a9392ee3e8226db1056f12b963e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:65.521ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\left(g(x)^{h(x)}\right)=h(x)g(x)^{h(x)-1}{\frac {d}{dx}}g(x)+g(x)^{h(x)}\ln g(x)\,{\frac {d}{dx}}h(x).}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="General_rule:_Vector-valued_functions_with_multiple_inputs">General rule: Vector-valued functions with multiple inputs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=18" title="Edit section: General rule: Vector-valued functions with multiple inputs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest way for writing the chain rule in the general case is to use the <a href="/wiki/Total_derivative#The_total_derivative_as_a_linear_map" title="Total derivative">total derivative</a>, which is a linear transformation that captures all <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivatives</a> in a single formula. Consider differentiable functions <span class="texhtml"><i>f</i> : <b>R</b><sup><i>m</i></sup> → <b>R</b><sup><i>k</i></sup></span> and <span class="texhtml"><i>g</i> : <b>R</b><sup><i>n</i></sup> → <b>R</b><sup><i>m</i></sup></span>, and a point <span class="texhtml"><b>a</b></span> in <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. Let <span class="texhtml"><i>D</i><sub><b>a</b></sub> <i>g</i></span> denote the total derivative of <span class="texhtml"><i>g</i></span> at <span class="texhtml"><b>a</b></span> and <span class="texhtml"><i>D</i><sub><i>g</i>(<b>a</b>)</sub> <i>f</i></span> denote the total derivative of <span class="texhtml"><i>f</i></span> at <span class="texhtml"><i>g</i>(<b>a</b>)</span>. These two derivatives are linear transformations <span class="texhtml"><b>R</b><sup><i>n</i></sup> → <b>R</b><sup><i>m</i></sup></span> and <span class="texhtml"><b>R</b><sup><i>m</i></sup> → <b>R</b><sup><i>k</i></sup></span>, respectively, so they can be composed. The chain rule for total derivatives is that their composite is the total derivative of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> at <span class="texhtml"><b>a</b></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mathbf {a} }(f\circ g)=D_{g(\mathbf {a} )}f\circ D_{\mathbf {a} }g,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mi>f</mi> <mo>∘</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> </msub> <mi>g</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mathbf {a} }(f\circ g)=D_{g(\mathbf {a} )}f\circ D_{\mathbf {a} }g,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddba5ff8b1f452d78d8987d06d766bfb0797c43d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.028ex; height:3.176ex;" alt="{\displaystyle D_{\mathbf {a} }(f\circ g)=D_{g(\mathbf {a} )}f\circ D_{\mathbf {a} }g,}" /></span> or for short, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(f\circ g)=Df\circ Dg.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>D</mi> <mi>f</mi> <mo>∘</mo> <mi>D</mi> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(f\circ g)=Df\circ Dg.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0f133ce31911e43c764654f1413b98c911dc7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.506ex; height:2.843ex;" alt="{\displaystyle D(f\circ g)=Df\circ Dg.}" /></span> The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.<sup id="cite_ref-spivak_manifolds_7-0" class="reference"><a href="#cite_note-spivak_manifolds-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Because the total derivative is a linear transformation, the functions appearing in the formula can be rewritten as matrices. The matrix corresponding to a total derivative is called a <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>, and the composite of two derivatives corresponds to the product of their Jacobian matrices. From this perspective the chain rule therefore says: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f\circ g}(\mathbf {a} )=J_{f}(g(\mathbf {a} ))J_{g}(\mathbf {a} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{f\circ g}(\mathbf {a} )=J_{f}(g(\mathbf {a} ))J_{g}(\mathbf {a} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1b5731ed718474bd9d8fa61241cad7e0c7337a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.773ex; height:3.009ex;" alt="{\displaystyle J_{f\circ g}(\mathbf {a} )=J_{f}(g(\mathbf {a} ))J_{g}(\mathbf {a} ),}" /></span> or for short, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{f\circ g}=(J_{f}\circ g)J_{g}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{f\circ g}=(J_{f}\circ g)J_{g}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68747370032dd147a77883a3af32fc57a46cfcb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.641ex; height:3.009ex;" alt="{\displaystyle J_{f\circ g}=(J_{f}\circ g)J_{g}.}" /></span> </p><p>That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points). </p><p>The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If <span class="texhtml mvar" style="font-style:italic;">k</span>, <span class="texhtml mvar" style="font-style:italic;">m</span>, and <span class="texhtml mvar" style="font-style:italic;">n</span> are 1, so that <span class="texhtml"><i>f</i> : <b>R</b> → <b>R</b></span> and <span class="texhtml"><i>g</i> : <b>R</b> → <b>R</b></span>, then the Jacobian matrices of <span class="texhtml"><i>f</i></span> and <span class="texhtml"><i>g</i></span> are <span class="texhtml">1 × 1</span>. Specifically, they are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J_{g}(a)&={\begin{pmatrix}g'(a)\end{pmatrix}},\\J_{f}(g(a))&={\begin{pmatrix}f'(g(a))\end{pmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{g}(a)&={\begin{pmatrix}g'(a)\end{pmatrix}},\\J_{f}(g(a))&={\begin{pmatrix}f'(g(a))\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7bca38d49cb8ae44c24b20583c09cd9f1f21c28" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.418ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}J_{g}(a)&={\begin{pmatrix}g'(a)\end{pmatrix}},\\J_{f}(g(a))&={\begin{pmatrix}f'(g(a))\end{pmatrix}}.\end{aligned}}}" /></span> The Jacobian of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> is the product of these <span class="texhtml">1 × 1</span> matrices, so it is <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))⋅<i>g</i>′(<i>a</i>)</span>, as expected from the one-dimensional chain rule. In the language of linear transformations, <span class="texhtml"><i>D</i><sub><i>a</i></sub>(<i>g</i>)</span> is the function which scales a vector by a factor of <span class="texhtml"><i>g</i>′(<i>a</i>)</span> and <span class="texhtml"><i>D</i><sub><i>g</i>(<i>a</i>)</sub>(<i>f</i>)</span> is the function which scales a vector by a factor of <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))</span>. The chain rule says that the composite of these two linear transformations is the linear transformation <span class="texhtml"><i>D</i><sub><i>a</i></sub>(<i>f</i> ∘ <i>g</i>)</span>, and therefore it is the function that scales a vector by <span class="texhtml"><i>f</i>′(<i>g</i>(<i>a</i>))⋅<i>g</i>′(<i>a</i>)</span>. </p><p>Another way of writing the chain rule is used when <i>f</i> and <i>g</i> are expressed in terms of their components as <span class="texhtml"><b>y</b> = <i>f</i>(<b>u</b>) = (<i>f</i><sub>1</sub>(<b>u</b>), …, <i>f</i><sub><i>k</i></sub>(<b>u</b>))</span> and <span class="texhtml"><b>u</b> = <i>g</i>(<b>x</b>) = (<i>g</i><sub>1</sub>(<b>x</b>), …, <i>g</i><sub><i>m</i></sub>(<b>x</b>))</span>. In this case, the above rule for Jacobian matrices is usually written as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (x_{1},\ldots ,x_{n})}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial (x_{1},\ldots ,x_{n})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (x_{1},\ldots ,x_{n})}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial (x_{1},\ldots ,x_{n})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ae9e07f60b83ed3a8c08e45f8f535c415cb43c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.88ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (x_{1},\ldots ,x_{n})}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial (x_{1},\ldots ,x_{n})}}.}" /></span> </p><p>The chain rule for total derivatives implies a chain rule for partial derivatives. Recall that when the total derivative exists, the partial derivative in the <span class="texhtml mvar" style="font-style:italic;">i</span>-th coordinate direction is found by multiplying the Jacobian matrix by the <span class="texhtml mvar" style="font-style:italic;">i</span>-th basis vector. By doing this to the formula above, we find: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial x_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial x_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcfc8c7770bcd35a22c4523f5ff9d0017ab6db2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.369ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial x_{i}}}.}" /></span> Since the entries of the Jacobian matrix are partial derivatives, we may simplify the above formula to get: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial (y_{1},\ldots ,y_{k})}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial (y_{1},\ldots ,y_{k})}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f89d5d763e618470d85c726cfdf3a6bdbb606d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.015ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial (y_{1},\ldots ,y_{k})}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.}" /></span> More conceptually, this rule expresses the fact that a change in the <span class="texhtml"><i>x</i><sub><i>i</i></sub></span> direction may change all of <span class="texhtml"><i>g</i><sub>1</sub></span> through <span class="texhtml"><i>g<sub>m</sub></i></span>, and any of these changes may affect <span class="texhtml"><i>f</i></span>. </p><p>In the special case where <span class="texhtml"><i>k</i> = 1</span>, so that <span class="texhtml"><i>f</i></span> is a real-valued function, then this formula simplifies even further: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial y}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial y}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d059d8743b6dc8824e042fa091e84c39d7db49c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.575ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial y}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial y}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.}" /></span> This can be rewritten as a <a href="/wiki/Dot_product" title="Dot product">dot product</a>. Recalling that <span class="texhtml"><b>u</b> = (<i>g</i><sub>1</sub>, …, <i>g</i><sub><i>m</i></sub>)</span>, the partial derivative <span class="texhtml">∂<b>u</b> / ∂<i>x</i><sub><i>i</i></sub></span> is also a vector, and the chain rule says that: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial x_{i}}}=\nabla y\cdot {\frac {\partial \mathbf {u} }{\partial x_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>y</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial x_{i}}}=\nabla y\cdot {\frac {\partial \mathbf {u} }{\partial x_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865553d07f871801f11a44e6892903a40ce1e3fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.083ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial y}{\partial x_{i}}}=\nabla y\cdot {\frac {\partial \mathbf {u} }{\partial x_{i}}}.}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=19" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given <span class="texhtml"><i>u</i>(<i>x</i>, <i>y</i>) = <i>x</i><sup>2</sup> + 2<i>y</i></span> where <span class="texhtml"><i>x</i>(<i>r</i>, <i>t</i>) = <i>r</i> sin(<i>t</i>)</span> and <span class="texhtml"><i>y</i>(<i>r</i>,<i>t</i>) = sin<sup>2</sup>(<i>t</i>)</span>, determine the value of <span class="texhtml">∂<i>u</i> / ∂<i>r</i></span> and <span class="texhtml">∂<i>u</i> / ∂<i>t</i></span> using the chain rule.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2023)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial r}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}}=(2x)(\sin(t))+(2)(0)=2r\sin ^{2}(t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>r</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial r}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}}=(2x)(\sin(t))+(2)(0)=2r\sin ^{2}(t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94bc190751446909028bb82cb1ac778adb888be3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.584ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial u}{\partial r}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}}=(2x)(\sin(t))+(2)(0)=2r\sin ^{2}(t),}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}&={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial t}}\\&=(2x)(r\cos(t))+(2)(2\sin(t)\cos(t))\\&=(2r\sin(t))(r\cos(t))+4\sin(t)\cos(t)\\&=2(r^{2}+2)\sin(t)\cos(t)\\&=(r^{2}+2)\sin(2t).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}&={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial t}}\\&=(2x)(r\cos(t))+(2)(2\sin(t)\cos(t))\\&=(2r\sin(t))(r\cos(t))+4\sin(t)\cos(t)\\&=2(r^{2}+2)\sin(t)\cos(t)\\&=(r^{2}+2)\sin(2t).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b60404f728a4abb92158e6bd81d65947fa9098a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:42.293ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}&={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial t}}\\&=(2x)(r\cos(t))+(2)(2\sin(t)\cos(t))\\&=(2r\sin(t))(r\cos(t))+4\sin(t)\cos(t)\\&=2(r^{2}+2)\sin(t)\cos(t)\\&=(r^{2}+2)\sin(2t).\end{aligned}}}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="Higher_derivatives_of_multivariable_functions">Higher derivatives of multivariable functions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=20" title="Edit section: Higher derivatives of multivariable functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula#Multivariate_version" title="Faà di Bruno's formula">Faà di Bruno's formula § Multivariate version</a></div> <p>Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case. If <span class="texhtml"><i>y</i> = <i>f</i>(<b>u</b>)</span> is a function of <span class="texhtml"><b>u</b> = <i>g</i>(<b>x</b>)</span> as above, then the second derivative of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}y}{\partial x_{i}\partial x_{j}}}=\sum _{k}\left({\frac {\partial y}{\partial u_{k}}}{\frac {\partial ^{2}u_{k}}{\partial x_{i}\partial x_{j}}}\right)+\sum _{k,\ell }\left({\frac {\partial ^{2}y}{\partial u_{k}\partial u_{\ell }}}{\frac {\partial u_{k}}{\partial x_{i}}}{\frac {\partial u_{\ell }}{\partial x_{j}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}y}{\partial x_{i}\partial x_{j}}}=\sum _{k}\left({\frac {\partial y}{\partial u_{k}}}{\frac {\partial ^{2}u_{k}}{\partial x_{i}\partial x_{j}}}\right)+\sum _{k,\ell }\left({\frac {\partial ^{2}y}{\partial u_{k}\partial u_{\ell }}}{\frac {\partial u_{k}}{\partial x_{i}}}{\frac {\partial u_{\ell }}{\partial x_{j}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb795344ae507800bc25a112f4fd902b51342e7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:58.667ex; height:7.176ex;" alt="{\displaystyle {\frac {\partial ^{2}y}{\partial x_{i}\partial x_{j}}}=\sum _{k}\left({\frac {\partial y}{\partial u_{k}}}{\frac {\partial ^{2}u_{k}}{\partial x_{i}\partial x_{j}}}\right)+\sum _{k,\ell }\left({\frac {\partial ^{2}y}{\partial u_{k}\partial u_{\ell }}}{\frac {\partial u_{k}}{\partial x_{i}}}{\frac {\partial u_{\ell }}{\partial x_{j}}}\right).}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Further_generalizations">Further generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=21" title="Edit section: Further generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All extensions of calculus have a chain rule. In most of these, the formula remains the same, though the meaning of that formula may be vastly different. </p><p>One generalization is to <a href="/wiki/Manifold" title="Manifold">manifolds</a>. In this situation, the chain rule represents the fact that the derivative of <span class="texhtml"><i>f</i> ∘ <i>g</i></span> is the composite of the derivative of <span class="texhtml"><i>f</i></span> and the derivative of <span class="texhtml"><i>g</i></span>. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. </p><p>The chain rule is also valid for <a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivatives</a> in <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>. The same formula holds as before.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This case and the previous one admit a simultaneous generalization to <a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifolds</a>. </p><p>In <a href="/wiki/Differential_algebra" title="Differential algebra">differential algebra</a>, the derivative is interpreted as a morphism of modules of <a href="/wiki/K%C3%A4hler_differential" title="Kähler differential">Kähler differentials</a>. A <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> of <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a> <span class="texhtml"><i>f</i> : <i>R</i> → <i>S</i></span> determines a morphism of Kähler differentials <span class="texhtml"><i>Df</i> : Ω<sub><i>R</i></sub> → Ω<sub><i>S</i></sub></span> which sends an element <span class="texhtml"><i>dr</i></span> to <span class="texhtml"><i>d</i>(<i>f</i>(<i>r</i>))</span>, the exterior differential of <span class="texhtml"><i>f</i>(<i>r</i>)</span>. The formula <span class="texhtml"><i>D</i>(<i>f</i> ∘ <i>g</i>) = <i>Df</i> ∘ <i>Dg</i></span> holds in this context as well. </p><p>The common feature of these examples is that they are expressions of the idea that the derivative is part of a <a href="/wiki/Functor" title="Functor">functor</a>. A functor is an operation on spaces and functions between them. It associates to each space a new space and to each function between two spaces a new function between the corresponding new spaces. In each of the above cases, the functor sends each space to its <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> and it sends each function to its derivative. For example, in the manifold case, the derivative sends a <span class="texhtml"><i>C</i><sup><i>r</i></sup></span>-manifold to a <span class="texhtml"><i>C</i><sup><i>r</i>−1</sup></span>-manifold (its tangent bundle) and a <span class="texhtml"><i>C</i><sup><i>r</i></sup></span>-function to its total derivative. There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. This is exactly the formula <span class="texhtml"><i>D</i>(<i>f</i> ∘ <i>g</i>) = <i>Df</i> ∘ <i>Dg</i></span>. </p><p>There are also chain rules in <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a>. One of these, <a href="/wiki/It%C5%8D%27s_lemma" class="mw-redirect" title="Itō's lemma">Itō's lemma</a>, expresses the composite of an Itō process (or more generally a <a href="/wiki/Semimartingale" title="Semimartingale">semimartingale</a>) <i>dX</i><sub><i>t</i></sub> with a twice-differentiable function <i>f</i>. In Itō's lemma, the derivative of the composite function depends not only on <i>dX</i><sub><i>t</i></sub> and the derivative of <i>f</i> but also on the second derivative of <i>f</i>. The dependence on the second derivative is a consequence of the non-zero <a href="/wiki/Quadratic_variation" title="Quadratic variation">quadratic variation</a> of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way. This variant of the chain rule is not an example of a functor because the two functions being composed are of different types. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=22" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Automatic_differentiation" title="Automatic differentiation">Automatic differentiation</a> – Numerical calculations carrying along derivatives − a computational method that makes heavy use of the chain rule to compute exact numerical derivatives.</li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Differentiation rules</a> – Rules for computing derivatives of functions</li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> – Technique in integral evaluation</li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a> – Differentiation under the integral sign formula</li> <li><a href="/wiki/Product_rule" title="Product rule">Product rule</a> – Formula for the derivative of a product</li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient rule</a> – Formula for the derivative of a ratio of functions</li> <li><a href="/wiki/Triple_product_rule" title="Triple product rule">Triple product rule</a> – Relation between relative derivatives of three variables</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="/wiki/George_F._Simmons" title="George F. Simmons">George F. Simmons</a>, <i>Calculus with Analytic Geometry</i> (1985), p. 93.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChild1917" class="citation journal cs1">Child, J. M. (1917). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27900650">"THE MANUSCRIPTS OF LEIBNIZ ON HIS DISCOVERY OF THE DIFFERENTIAL CALCULUS. PART II (Continued)"</a>. <i>The Monist</i>. <b>27</b> (3): <span class="nowrap">411–</span>454. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5840%2Fmonist191727324">10.5840/monist191727324</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-9662">0026-9662</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27900650">27900650</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Monist&rft.atitle=THE+MANUSCRIPTS+OF+LEIBNIZ+ON+HIS+DISCOVERY+OF+THE+DIFFERENTIAL+CALCULUS.+PART+II+%28Continued%29&rft.volume=27&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E411-%3C%2Fspan%3E454&rft.date=1917&rft.issn=0026-9662&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27900650%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.5840%2Fmonist191727324&rft.aulast=Child&rft.aufirst=J.+M.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27900650&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span> </li> <li id="cite_note-OHR-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-OHR_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-OHR_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRodríguezLópez_Fernández2010" class="citation journal cs1">Rodríguez, Omar Hernández; López Fernández, Jorge M. (2010). <a rel="nofollow" class="external text" href="https://scholarworks.umt.edu/tme/vol7/iss2/10/">"A Semiotic Reflection on the Didactics of the Chain Rule"</a>. <i>The Mathematics Enthusiast</i>. <b>7</b> (2): <span class="nowrap">321–</span>332. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.54870%2F1551-3440.1191">10.54870/1551-3440.1191</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29739148">29739148</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematics+Enthusiast&rft.atitle=A+Semiotic+Reflection+on+the+Didactics+of+the+Chain+Rule&rft.volume=7&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E321-%3C%2Fspan%3E332&rft.date=2010&rft_id=info%3Adoi%2F10.54870%2F1551-3440.1191&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A29739148%23id-name%3DS2CID&rft.aulast=Rodr%C3%ADguez&rft.aufirst=Omar+Hern%C3%A1ndez&rft.au=L%C3%B3pez+Fern%C3%A1ndez%2C+Jorge+M.&rft_id=https%3A%2F%2Fscholarworks.umt.edu%2Ftme%2Fvol7%2Fiss2%2F10%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFApostol1974" class="citation book cs1"><a href="/wiki/Tom_Apostol" class="mw-redirect" title="Tom Apostol">Apostol, Tom</a> (1974). <i>Mathematical analysis</i> (2nd ed.). Addison Wesley. Theorem 5.5.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+analysis&rft.pages=Theorem+5.5&rft.edition=2nd&rft.pub=Addison+Wesley&rft.date=1974&rft.aulast=Apostol&rft.aufirst=Tom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGoodfellowBengioCourville2016" class="citation cs2"><a href="/wiki/Ian_Goodfellow" title="Ian Goodfellow">Goodfellow, Ian</a>; <a href="/wiki/Yoshua_Bengio" title="Yoshua Bengio">Bengio, Yoshua</a>; Courville, Aaron (2016), <i>Deep learning</i>, MIT</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deep+learning&rft.pub=MIT&rft.date=2016&rft.aulast=Goodfellow&rft.aufirst=Ian&rft.au=Bengio%2C+Yoshua&rft.au=Courville%2C+Aaron&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span>, pp=197–217.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKuhn1991" class="citation journal cs1">Kuhn, Stephen (1991). "The Derivative á la Carathéodory". <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>98</b> (1): <span class="nowrap">40–</span>44. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2324035">10.2307/2324035</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2324035">2324035</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Derivative+%C3%A1+la+Carath%C3%A9odory&rft.volume=98&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E40-%3C%2Fspan%3E44&rft.date=1991&rft_id=info%3Adoi%2F10.2307%2F2324035&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2324035%23id-name%3DJSTOR&rft.aulast=Kuhn&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span> </li> <li id="cite_note-spivak_manifolds-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-spivak_manifolds_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSpivak1965" class="citation book cs1"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1965). <i><a href="/wiki/Calculus_on_Manifolds_(book)" title="Calculus on Manifolds (book)">Calculus on Manifolds</a></i>. Boston: Addison-Wesley. pp. <span class="nowrap">19–</span>20. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-9021-9" title="Special:BookSources/0-8053-9021-9"><bdi>0-8053-9021-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+on+Manifolds&rft.place=Boston&rft.pages=%3Cspan+class%3D%22nowrap%22%3E19-%3C%2Fspan%3E20&rft.pub=Addison-Wesley&rft.date=1965&rft.isbn=0-8053-9021-9&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCheney2001" class="citation book cs1"><a href="/wiki/Elliott_Ward_Cheney_Jr." title="Elliott Ward Cheney Jr.">Cheney, Ward</a> (2001). 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New York: Springer. pp. <span class="nowrap">121–</span>125. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95279-9" title="Special:BookSources/0-387-95279-9"><bdi>0-387-95279-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Chain+Rule+and+Mean+Value+Theorems&rft.btitle=Analysis+for+Applied+Mathematics&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E121-%3C%2Fspan%3E125&rft.pub=Springer&rft.date=2001&rft.isbn=0-387-95279-9&rft.aulast=Cheney&rft.aufirst=Ward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chain_rule&action=edit&section=24" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Leibniz_rule">"Leibniz rule"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Leibniz+rule&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLeibniz_rule&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Chain_Rule"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ChainRule.html">"Chain Rule"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Chain+Rule&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FChainRule.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChain+rule" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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class="navbox-group" style="width:1%"><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li> <li><a href="/wiki/Concave_function" title="Concave function">Concave function</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous function</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a class="mw-selflink selflink">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: 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Chain rule - Wikipedia