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class="vector-toc-numb">2</span> <span>Tangent line to a plane curve</span> </div> </a> <button aria-controls="toc-Tangent_line_to_a_plane_curve-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 Tangent line to a plane curve subsection</span> </button> <ul id="toc-Tangent_line_to_a_plane_curve-sublist" class="vector-toc-list"> <li id="toc-Analytical_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytical_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Analytical approach</span> </div> </a> <ul id="toc-Analytical_approach-sublist" class="vector-toc-list"> <li id="toc-Intuitive_description" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Intuitive_description"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Intuitive description</span> </div> </a> <ul id="toc-Intuitive_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-More_rigorous_description" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#More_rigorous_description"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>More rigorous description</span> </div> </a> <ul id="toc-More_rigorous_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-How_the_method_can_fail" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#How_the_method_can_fail"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>How the method can fail</span> </div> </a> <ul id="toc-How_the_method_can_fail-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Equations</span> </div> </a> <ul id="toc-Equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normal_line_to_a_curve" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_line_to_a_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Normal line to a curve</span> </div> </a> <ul id="toc-Normal_line_to_a_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Angle_between_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angle_between_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Angle between curves</span> </div> </a> <ul id="toc-Angle_between_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple_tangents_at_a_point" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_tangents_at_a_point"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Multiple tangents at a point</span> </div> </a> <ul id="toc-Multiple_tangents_at_a_point-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tangent_line_to_a_space_curve" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tangent_line_to_a_space_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Tangent line to a space curve</span> </div> </a> <ul id="toc-Tangent_line_to_a_space_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangent_circles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tangent_circles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Tangent circles</span> </div> </a> <ul id="toc-Tangent_circles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangent_plane_to_a_surface" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tangent_plane_to_a_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Tangent plane to a surface</span> </div> </a> <ul id="toc-Tangent_plane_to_a_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-dimensional_manifolds" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-dimensional_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Higher-dimensional manifolds</span> </div> </a> <ul id="toc-Higher-dimensional_manifolds-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">7</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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-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">10</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">Tangent</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 65 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-65" 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">65 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%85%D8%A7%D8%B3" title="مماس – Arabic" lang="ar" hreflang="ar" data-title="مماس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tanxente" title="Tanxente – Asturian" lang="ast" hreflang="ast" data-title="Tanxente" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%AA%E0%A6%B0%E0%A7%8D%E0%A6%B6%E0%A6%95" title="স্পর্শক – Bangla" lang="bn" hreflang="bn" data-title="স্পর্শক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%BE%D0%BF%D0%B8%D1%80%D0%B0%D1%82%D0%B5%D0%BB%D0%BD%D0%B0" title="Допирателна – Bulgarian" lang="bg" hreflang="bg" data-title="Допирателна" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tangent" title="Tangent – Catalan" lang="ca" hreflang="ca" data-title="Tangent" 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%A1%C4%95%D1%80%D1%82%C4%95%D0%BD%D0%B5%D0%B2%C4%95%D1%88" title="Сĕртĕневĕш – Chuvash" lang="cv" hreflang="cv" data-title="Сĕртĕневĕш" 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/Te%C4%8Dna" title="Tečna – Czech" lang="cs" hreflang="cs" data-title="Tečna" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Tangiad" title="Tangiad – Welsh" lang="cy" hreflang="cy" data-title="Tangiad" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tangent_(geometri)" title="Tangent (geometri) – Danish" lang="da" hreflang="da" data-title="Tangent (geometri)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Tangente" title="Tangente – German" lang="de" hreflang="de" data-title="Tangente" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Puutuja" title="Puutuja – Estonian" lang="et" hreflang="et" data-title="Puutuja" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tangente_(geometr%C3%ADa)" title="Tangente (geometría) – Spanish" lang="es" hreflang="es" data-title="Tangente (geometría)" 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/Tan%C4%9Danto" title="Tanĝanto – Esperanto" lang="eo" hreflang="eo" data-title="Tanĝanto" 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/Zuzen_ukitzaile" title="Zuzen ukitzaile – Basque" lang="eu" hreflang="eu" data-title="Zuzen ukitzaile" 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%85%D9%85%D8%A7%D8%B3" title="مماس – Persian" lang="fa" hreflang="fa" data-title="مماس" 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/Tangente_(g%C3%A9om%C3%A9trie)" title="Tangente (géométrie) – French" lang="fr" hreflang="fr" data-title="Tangente (géométrie)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tadhla%C3%AD" title="Tadhlaí – Irish" lang="ga" hreflang="ga" data-title="Tadhlaí" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tanxente" title="Tanxente – Galician" lang="gl" hreflang="gl" data-title="Tanxente" 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%A0%91%EC%84%A0" title="접선 – Korean" lang="ko" hreflang="ko" data-title="접선" 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/%D5%87%D5%B8%D5%B7%D5%A1%D6%83%D5%B8%D5%B2" title="Շոշափող – Armenian" lang="hy" hreflang="hy" data-title="Շոշափող" 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%B8%E0%A5%8D%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%B6%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE" title="स्पर्शरेखा – Hindi" lang="hi" hreflang="hi" data-title="स्पर्शरेखा" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Tangenta" title="Tangenta – Croatian" lang="hr" hreflang="hr" data-title="Tangenta" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Garis_singgung" title="Garis singgung – Indonesian" lang="id" hreflang="id" data-title="Garis singgung" 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/Snertill" title="Snertill – Icelandic" lang="is" hreflang="is" data-title="Snertill" 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/Tangente_(geometria)" title="Tangente (geometria) – Italian" lang="it" hreflang="it" data-title="Tangente (geometria)" 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%9E%D7%A9%D7%99%D7%A7" title="משיק – Hebrew" lang="he" hreflang="he" data-title="משיק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B0%D0%BD%D0%B0%D0%BC%D0%B0" title="Жанама – Kazakh" lang="kk" hreflang="kk" data-title="Жанама" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D0%BB%D0%B0%D0%BD%D0%B0%D0%B3%D0%B0_%D0%B6%D0%B0%D0%BD%D1%8B%D0%BC%D0%B0" title="Айланага жаныма – Kyrgyz" lang="ky" hreflang="ky" data-title="Айланага жаныма" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Liestin%C4%97" title="Liestinė – Lithuanian" lang="lt" hreflang="lt" data-title="Liestinė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Tangent" title="Tangent – Lombard" lang="lmo" hreflang="lmo" data-title="Tangent" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%89rint%C5%91_(k%C3%B6r)" title="Érintő (kör) – Hungarian" lang="hu" hreflang="hu" data-title="Érintő (kör)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%BE%D0%BF%D0%B8%D1%80%D0%BA%D0%B0" title="Допирка – Macedonian" lang="mk" hreflang="mk" data-title="Допирка" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8A%E0%B4%9F%E0%B5%81%E0%B4%B5%E0%B4%B0" title="തൊടുവര – Malayalam" lang="ml" hreflang="ml" data-title="തൊടുവര" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tangen" title="Tangen – Malay" lang="ms" hreflang="ms" data-title="Tangen" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Raaklijn" title="Raaklijn – Dutch" lang="nl" hreflang="nl" data-title="Raaklijn" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%B6_%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE" title="स्पर्श रेखा – Nepali" lang="ne" hreflang="ne" data-title="स्पर्श रेखा" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8E%A5%E7%B7%9A" title="接線 – Japanese" lang="ja" hreflang="ja" data-title="接線" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Tangent" title="Tangent – Northern Frisian" lang="frr" hreflang="frr" data-title="Tangent" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tangent_(matematikk)" title="Tangent (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Tangent (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tangent" title="Tangent – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tangent" 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/Tangenta" title="Tangenta – Piedmontese" lang="pms" hreflang="pms" data-title="Tangenta" 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/Styczna" title="Styczna – Polish" lang="pl" hreflang="pl" data-title="Styczna" 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/Tangente_(geometria)" title="Tangente (geometria) – Portuguese" lang="pt" hreflang="pt" data-title="Tangente (geometria)" 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/Tangent%C4%83_(geometrie)" title="Tangentă (geometrie) – Romanian" lang="ro" hreflang="ro" data-title="Tangentă (geometrie)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Patan_siq%27i" title="Patan siq'i – Quechua" lang="qu" hreflang="qu" data-title="Patan siq'i" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D1%81%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D1%8F%D0%BC%D0%B0%D1%8F" title="Касательная прямая – Russian" lang="ru" hreflang="ru" data-title="Касательная прямая" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Tangentja" title="Tangentja – Albanian" lang="sq" hreflang="sq" data-title="Tangentja" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tangent_(geometry)" title="Tangent (geometry) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tangent (geometry)" 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/Tangenta" title="Tangenta – Slovenian" lang="sl" hreflang="sl" data-title="Tangenta" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Tangent" title="Tangent – Somali" lang="so" hreflang="so" data-title="Tangent" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%8E%DA%A9%DB%95%D9%88%D8%AA" title="لێکەوت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لێکەوت" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BD%D0%B3%D0%B5%D0%BD%D1%82%D0%B0" title="Тангента – Serbian" lang="sr" hreflang="sr" data-title="Тангента" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tangenta" title="Tangenta – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Tangenta" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tangentti_(geometria)" title="Tangentti (geometria) – Finnish" lang="fi" hreflang="fi" data-title="Tangentti (geometria)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tangent_(matematik)" title="Tangent (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Tangent (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tangent" title="Tangent – Tagalog" lang="tl" hreflang="tl" data-title="Tangent" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81" title="தொடுகோடு – Tamil" lang="ta" hreflang="ta" data-title="தொடுகோடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99%E0%B8%AA%E0%B8%B1%E0%B8%A1%E0%B8%9C%E0%B8%B1%E0%B8%AA" title="เส้นสัมผัส – Thai" lang="th" hreflang="th" data-title="เส้นสัมผัส" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Te%C4%9Fet" title="Teğet – Turkish" lang="tr" hreflang="tr" data-title="Teğet" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%BE%D1%82%D0%B8%D1%87%D0%BD%D0%B0" title="Дотична – Ukrainian" lang="uk" hreflang="uk" data-title="Дотична" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">In mathematics, straight line touching a plane curve without crossing it</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">For the tangent function, see <a href="/wiki/Tangent_(trigonometry)" class="mw-redirect" title="Tangent (trigonometry)">Tangent (trigonometry)</a>. For other uses, see <a href="/wiki/Tangent_(disambiguation)" class="mw-disambig" title="Tangent (disambiguation)">Tangent (disambiguation)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tangent_to_a_curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Tangent_to_a_curve.svg/220px-Tangent_to_a_curve.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Tangent_to_a_curve.svg/330px-Tangent_to_a_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Tangent_to_a_curve.svg/440px-Tangent_to_a_curve.svg.png 2x" data-file-width="400" data-file-height="280" /></a><figcaption>Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Image_Tangent-plane.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Image_Tangent-plane.svg/250px-Image_Tangent-plane.svg.png" decoding="async" width="220" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Image_Tangent-plane.svg/330px-Image_Tangent-plane.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Image_Tangent-plane.svg/440px-Image_Tangent-plane.svg.png 2x" data-file-width="440" data-file-height="283" /></a><figcaption>Tangent plane to a sphere</figcaption></figure> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the <b>tangent line</b> (or simply <b>tangent</b>) to a plane <a href="/wiki/Curve" title="Curve">curve</a> at a given <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> is, intuitively, the <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> that "just touches" the curve at that point. <a href="/wiki/Leibniz" class="mw-redirect" title="Leibniz">Leibniz</a> defined it as the line through a pair of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitely close</a> points on the curve.<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><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> More precisely, a straight line is tangent to the curve <span class="nowrap"><i>y</i> = <i>f</i>(<i>x</i>)</span> at a point <span class="nowrap"><i>x</i> = <i>c</i></span> if the line passes through the point <span class="nowrap">(<i>c</i>, <i>f</i>(<i>c</i>))</span> on the curve and has <a href="/wiki/Slope" title="Slope">slope</a> <span class="nowrap"><i>f</i><span class="nowrap" style="padding-left:0.1em;">'</span>(<i>c</i>)</span>, where <i>f</i><span class="nowrap" style="padding-left:0.1em;">'</span> is the <a href="/wiki/Derivative" title="Derivative">derivative</a> of <i>f</i>. A similar definition applies to <a href="/wiki/Space_curve" class="mw-redirect" title="Space curve">space curves</a> and curves in <i>n</i>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. </p><p>The point where the tangent line and the curve meet or <a href="/wiki/Intersection_(geometry)" title="Intersection (geometry)">intersect</a> is called the <i><b>point of tangency</b></i>. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a <i><a href="/wiki/Tangent_line_approximation" class="mw-redirect" title="Tangent line approximation">tangent line approximation</a></i>, the graph of the <a href="/wiki/Affine_function" class="mw-redirect" title="Affine function">affine function</a> that best approximates the original function at the given point.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Similarly, the <b>tangent plane</b> to a <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a> at a given point is the <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and has been extensively generalized; <style data-mw-deduplicate="TemplateStyles:r1033199720">.mw-parser-output div.crossreference{padding-left:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><span role="note" class="hatnote navigation-not-searchable crossreference">see <a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></span>. </p><p>The word "tangent" comes from the <a href="/wiki/Latin" title="Latin">Latin</a> <span title="Latin-language text"><i lang="la"><a href="https://en.wiktionary.org/wiki/en:tangere#Latin" class="extiw" title="wikt:en:tangere">tangere</a></i></span>, "to touch". </p> <meta property="mw:PageProp/toc" /> <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=Tangent&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Euclid" title="Euclid">Euclid</a> makes several references to the tangent (<span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἐφαπτομένη</span></span> <i>ephaptoménē</i>) to a circle in book III of the <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i> (c. 300 BC).<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> In <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a>' work <i>Conics</i> (c. 225 BC) he defines a tangent as being <i>a line such that no other straight line could fall between it and the curve</i>.<sup id="cite_ref-Shenk_5-0" class="reference"><a href="#cite_note-Shenk-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (c. 287 – c. 212 BC) found the tangent to an <a href="/wiki/Archimedean_spiral" title="Archimedean spiral">Archimedean spiral</a> by considering the path of a point moving along the curve.<sup id="cite_ref-Shenk_5-1" class="reference"><a href="#cite_note-Shenk-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In the 1630s <a href="/wiki/Fermat" class="mw-redirect" title="Fermat">Fermat</a> developed the technique of <a href="/wiki/Adequality" title="Adequality">adequality</a> to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adequality is similar to taking the difference between <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(x+h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79d112f0e80b791b42f289ae9b9b520ca2edd28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.597ex; height:2.843ex;" alt="{\displaystyle f(x+h)}" /></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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" /></span> and dividing by a power 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /></span>. Independently <a href="/wiki/Descartes" class="mw-redirect" title="Descartes">Descartes</a> used his <a href="/wiki/Method_of_normals" title="Method of normals">method of normals</a> based on the observation that the radius of a circle is always normal to the circle itself.<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>These methods led to the development of <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> in the 17th century. Many people contributed. <a href="/wiki/Gilles_de_Roberval" title="Gilles de Roberval">Roberval</a> discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Ren%C3%A9-Fran%C3%A7ois_de_Sluse" title="René-François de Sluse">René-François de Sluse</a> and <a href="/wiki/Johannes_Hudde" title="Johannes Hudde">Johannes Hudde</a> found algebraic algorithms for finding tangents.<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> Further developments included those of <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> and <a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a>, leading to the theory of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a>. </p><p>An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This old definition prevents <a href="/wiki/Inflection_point" title="Inflection point">inflection points</a> from having any tangent. It has been dismissed and the modern definitions are equivalent to those of <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>, who defined the tangent line as the line through a pair of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitely close</a> points on the curve; in modern terminology, this is expressed as: the tangent to a curve at a point <span class="texhtml mvar" style="font-style:italic;">P</span> on the curve is the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of the line passing through two points of the curve when these two points tends to <span class="texhtml mvar" style="font-style:italic;">P</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Tangent_line_to_a_plane_curve">Tangent line to a plane curve<span class="anchor" id="Line"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=2" title="Edit section: Tangent line to a plane curve"><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">Further information: <a href="/wiki/Differentiable_curve#Tangent_vector" title="Differentiable curve">Differentiable curve § Tangent vector</a>, and <a href="/wiki/Frenet%E2%80%93Serret_formulas" title="Frenet–Serret formulas">Frenet–Serret formulas</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:CIRCLE_LINES-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/220px-CIRCLE_LINES-en.svg.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/330px-CIRCLE_LINES-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/440px-CIRCLE_LINES-en.svg.png 2x" data-file-width="612" data-file-height="618" /></a><figcaption>A tangent, a <a href="/wiki/Chord_(geometry)" title="Chord (geometry)">chord</a>, and a <a href="/wiki/Secant_line" title="Secant line">secant</a> to a circle</figcaption></figure> <p>The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (<a href="/wiki/Secant_line" title="Secant line">secant lines</a>) passing through two points, <i>A</i> and <i>B</i>, those that lie on the function curve. The tangent at <i>A</i> is the limit when point <i>B</i> approximates or tends to <i>A</i>. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point <i>B</i>" approaches the vertex. </p><p>At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an <i><a href="/wiki/Inflection_point" title="Inflection point">inflection point</a></i>. <a href="/wiki/Circle" title="Circle">Circles</a>, <a href="/wiki/Parabola" title="Parabola">parabolas</a>, <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolas</a> and <a href="/wiki/Ellipse" title="Ellipse">ellipses</a> do not have any inflection point, but more complicated curves do have, like the graph of a <a href="/wiki/Cubic_function" title="Cubic function">cubic function</a>, which has exactly one inflection point, or a sinusoid, which has two inflection points per each <a href="/wiki/Periodic_function" title="Periodic function">period</a> of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a>. </p><p>Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a <a href="/wiki/Triangle" title="Triangle">triangle</a> and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In <a href="/wiki/Convex_geometry" title="Convex geometry">convex geometry</a>, such lines are called <a href="/wiki/Supporting_hyperplane" title="Supporting hyperplane">supporting lines</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Analytical_approach">Analytical approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=3" title="Edit section: Analytical approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Graph_of_sliding_derivative_line.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Graph_of_sliding_derivative_line.gif/330px-Graph_of_sliding_derivative_line.gif" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/7a/Graph_of_sliding_derivative_line.gif 1.5x" data-file-width="400" data-file-height="400" /></a><figcaption>At each point, the moving line is always tangent to the <a href="/wiki/Curve" title="Curve">curve</a>. Its slope is the <a href="/wiki/Derivative" title="Derivative">derivative</a>; green marks positive derivative, red marks negative derivative and black marks zero derivative. The point (x,y) = (0,1) where the tangent intersects the curve, is not a <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">max</a>, or a min, but is a <a href="/wiki/Point_of_inflection" class="mw-redirect" title="Point of inflection">point of inflection</a>. (Note: the figure contains the incorrect labeling of 0,0 which should be 0,1)</figcaption></figure> <p>The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the <b>tangent line problem,</b> was one of the central questions leading to the development of <a href="/wiki/Calculus" title="Calculus">calculus</a> in the 17th century. In the second book of his <i><a href="/wiki/La_Geometrie" class="mw-redirect" title="La Geometrie">Geometry</a></i>, <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="https://en.wikisource.org/wiki/fr:Page:Descartes_La_G%C3%A9om%C3%A9trie.djvu/52" class="extiw" title="s:fr:Page:Descartes La Géométrie.djvu/52">said</a> of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Intuitive_description">Intuitive description</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=4" title="Edit section: Intuitive description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that a curve is given as the graph of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, <i>y</i> = <i>f</i>(<i>x</i>). To find the tangent line at the point <i>p</i> = (<i>a</i>, <i>f</i>(<i>a</i>)), consider another nearby point <i>q</i> = (<i>a</i> + <i>h</i>, <i>f</i>(<i>a</i> + <i>h</i>)) on the curve. The <a href="/wiki/Slope" title="Slope">slope</a> of the <a href="/wiki/Secant_line" title="Secant line">secant line</a> passing through <i>p</i> and <i>q</i> is equal to the <a href="/wiki/Difference_quotient" title="Difference quotient">difference quotient</a> </p><p><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 {f(a+h)-f(a)}{h}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c924ec6626c572300b256a28bbb285cc98fe040" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.138ex; height:5.843ex;" alt="{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}" /></span> </p><p>As the point <i>q</i> approaches <i>p</i>, which corresponds to making <i>h</i> smaller and smaller, the difference quotient should approach a certain limiting value <i>k</i>, which is the slope of the tangent line at the point <i>p</i>. If <i>k</i> is known, the equation of the tangent line can be found in the point-slope form: </p><p><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-f(a)=k(x-a).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-f(a)=k(x-a).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c724e4b5c789c601569f0da3a65454294b44b79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.866ex; height:2.843ex;" alt="{\displaystyle y-f(a)=k(x-a).\,}" /></span> </p> <div class="mw-heading mw-heading4"><h4 id="More_rigorous_description">More rigorous description</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=5" title="Edit section: More rigorous description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value <i>k</i>. The precise mathematical formulation was given by <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a> in the 19th century and is based on the notion of <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a>. Suppose that the graph does not have a break or a sharp edge at <i>p</i> and it is neither plumb nor too wiggly near <i>p</i>. Then there is a unique value of <i>k</i> such that, as <i>h</i> approaches 0, the difference quotient gets closer and closer to <i>k</i>, and the distance between them becomes negligible compared with the size of <i>h</i>, if <i>h</i> is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function <i>f</i>. This limit is the <a href="/wiki/Derivative#Definition_via_difference_quotients" title="Derivative">derivative</a> of the function <i>f</i> at <i>x</i> = <i>a</i>, denoted <i>f</i> ′(<i>a</i>). Using derivatives, the equation of the tangent line can be stated as follows: </p> <dl><dd><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(a)+f'(a)(x-a).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</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>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(a)+f'(a)(x-a).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965437402e3a1f3ce00d7bd86c09b9b3ede10d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.7ex; height:3.009ex;" alt="{\displaystyle y=f(a)+f'(a)(x-a).\,}" /></span></dd></dl> <p>Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the <a href="/wiki/Power_function" class="mw-redirect" title="Power function">power function</a>, <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>, and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus. </p> <div class="mw-heading mw-heading4"><h4 id="How_the_method_can_fail">How the method can fail</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=6" title="Edit section: How the method can fail"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function <i>f</i> is <i>non-differentiable</i>. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent. </p><p>The graph <i>y</i> = <i>x</i><sup>1/3</sup> illustrates the first possibility: here the difference quotient at <i>a</i> = 0 is equal to <i>h</i><sup>1/3</sup>/<i>h</i> = <i>h</i><sup>−2/3</sup>, which becomes very large as <i>h</i> approaches 0. This curve has a tangent line at the origin that is vertical. </p><p>The graph <i>y</i> = <i>x</i><sup>2/3</sup> illustrates another possibility: this graph has a <i><a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">cusp</a></i> at the origin. This means that, when <i>h</i> approaches 0, the difference quotient at <i>a</i> = 0 approaches plus or minus infinity depending on the sign of <i>x</i>. Thus both branches of the curve are near to the half vertical line for which <i>y</i>=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, as a <i>double tangent</i>. </p><p>The graph <i>y</i> = |<i>x</i>| of the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> function consists of two straight lines with different slopes joined at the origin. As a point <i>q</i> approaches the origin from the right, the secant line always has slope 1. As a point <i>q</i> approaches the origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a <i>corner</i>. </p><p>Finally, since differentiability implies continuity, the <a href="/wiki/Contraposition" title="Contraposition">contrapositive</a> states <i>discontinuity</i> implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity </p> <div class="mw-heading mw-heading3"><h3 id="Equations">Equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=7" title="Edit section: Equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the curve is given by <i>y</i> = <i>f</i>(<i>x</i>) then the slope of the tangent 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 dy/dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy/dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbe4907fdff0ed999bb704dd0058cddd9f3e675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.726ex; height:2.843ex;" alt="{\displaystyle dy/dx,}" /></span> so by the <a href="/wiki/Linear_equation#Point–slope_form_or_Point-gradient_form" title="Linear equation">point–slope formula</a> the equation of the tangent line at (<i>X</i>, <i>Y</i>) is </p> <dl><dd><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-Y={\frac {dy}{dx}}(X)\cdot (x-X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <mi>Y</mi> <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 stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-Y={\frac {dy}{dx}}(X)\cdot (x-X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a790ded61bed61e14545c5ba37f9974f18298d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.677ex; height:5.509ex;" alt="{\displaystyle y-Y={\frac {dy}{dx}}(X)\cdot (x-X)}" /></span></dd></dl> <p>where (<i>x</i>, <i>y</i>) are the coordinates of any point on the tangent line, and where the derivative is evaluated at <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=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef78499ff6c3e72781859cfc41329b199b550bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.408ex; height:2.176ex;" alt="{\displaystyle x=X}" /></span>.<sup id="cite_ref-E191_12-0" class="reference"><a href="#cite_note-E191-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>When the curve is given by <i>y</i> = <i>f</i>(<i>x</i>), the tangent line's equation can also be found<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> by using <a href="/wiki/Polynomial_division" class="mw-redirect" title="Polynomial division">polynomial division</a> to divide <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\,(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad330d9fb9a55de9c53cf4d9180bd27a8b18e970" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f\,(x)}" /></span> by <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-X)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-X)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5999ba86ca976faf17a79cc19feae69c05cd3737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.014ex; height:3.176ex;" alt="{\displaystyle (x-X)^{2}}" /></span>; if the remainder is denoted by <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> </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/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" /></span>, then the equation of the tangent line is given by </p> <dl><dd><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=g(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=g(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a8ed7668a45c00356659eb684cfd430f479ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.156ex; height:2.843ex;" alt="{\displaystyle y=g(x).}" /></span></dd></dl> <p>When the equation of the curve is given in the form <i>f</i>(<i>x</i>, <i>y</i>) = 0 then the value of the slope can be found by <a href="/wiki/Implicit_and_explicit_functions#Implicit_differentiation" class="mw-redirect" title="Implicit and explicit functions">implicit differentiation</a>, giving </p> <dl><dd><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 {\frac {dy}{dx}}=-{\frac {\partial f}{\partial x}}{\bigg /}{\frac {\partial f}{\partial y}}.}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="2.047em" minsize="2.047em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=-{\frac {\partial f}{\partial x}}{\bigg /}{\frac {\partial f}{\partial y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93f34950e906b5b2b43497a6a5a74385369272e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.278ex; height:6.176ex;" alt="{\displaystyle {\frac {dy}{dx}}=-{\frac {\partial f}{\partial x}}{\bigg /}{\frac {\partial f}{\partial y}}.}" /></span></dd></dl> <p>The equation of the tangent line at a point (<i>X</i>,<i>Y</i>) such that <i>f</i>(<i>X</i>,<i>Y</i>) = 0 is then<sup id="cite_ref-E191_12-1" class="reference"><a href="#cite_note-E191-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\frac {\partial f}{\partial x}}(X,Y)\cdot (x-X)+{\frac {\partial f}{\partial y}}(X,Y)\cdot (y-Y)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot (x-X)+{\frac {\partial f}{\partial y}}(X,Y)\cdot (y-Y)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869ea1c2363776c2f2499591c106d169295c6209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.754ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot (x-X)+{\frac {\partial f}{\partial y}}(X,Y)\cdot (y-Y)=0.}" /></span></dd></dl> <p>This equation remains true if </p> <dl><dd><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 {\frac {\partial f}{\partial y}}(X,Y)=0,\quad {\frac {\partial f}{\partial x}}(X,Y)\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial y}}(X,Y)=0,\quad {\frac {\partial f}{\partial x}}(X,Y)\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b238aeb2b5e00016146be209da97a91c33c9c458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.635ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial f}{\partial y}}(X,Y)=0,\quad {\frac {\partial f}{\partial x}}(X,Y)\neq 0,}" /></span></dd></dl> <p>in which case the slope of the tangent is infinite. If, however, </p> <dl><dd><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 {\frac {\partial f}{\partial y}}(X,Y)={\frac {\partial f}{\partial x}}(X,Y)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial y}}(X,Y)={\frac {\partial f}{\partial x}}(X,Y)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d92bc10aa73f84311e5a7f7d709a129bce22d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.116ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial f}{\partial y}}(X,Y)={\frac {\partial f}{\partial x}}(X,Y)=0,}" /></span></dd></dl> <p>the tangent line is not defined and the point (<i>X</i>,<i>Y</i>) is said to be <a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">singular</a>. </p> <div style="clear:both;" class=""></div> <p>For <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a>, computations may be simplified somewhat by converting to <a href="/wiki/Homogeneous_coordinate" class="mw-redirect" title="Homogeneous coordinate">homogeneous coordinates</a>. Specifically, let the homogeneous equation of the curve be <i>g</i>(<i>x</i>, <i>y</i>, <i>z</i>) = 0 where <i>g</i> is a homogeneous function of degree <i>n</i>. Then, if (<i>X</i>, <i>Y</i>, <i>Z</i>) lies on the curve, <a href="/wiki/Homogeneous_function#Positive_homogeneity" title="Homogeneous function">Euler's theorem</a> implies <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 g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>X</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>Y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>Z</mi> <mo>=</mo> <mi>n</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d964b7e5be1f90a960e4f294b830debab2b030be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.043ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.}" /></span> It follows that the homogeneous equation of the tangent line is </p> <dl><dd><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 {\frac {\partial g}{\partial x}}(X,Y,Z)\cdot x+{\frac {\partial g}{\partial y}}(X,Y,Z)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,Z)\cdot z=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>z</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g}{\partial x}}(X,Y,Z)\cdot x+{\frac {\partial g}{\partial y}}(X,Y,Z)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,Z)\cdot z=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e96ae8bcac941962d26361b156108bd0651cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.196ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial g}{\partial x}}(X,Y,Z)\cdot x+{\frac {\partial g}{\partial y}}(X,Y,Z)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,Z)\cdot z=0.}" /></span></dd></dl> <p>The equation of the tangent line in Cartesian coordinates can be found by setting <i>z</i>=1 in this equation.<sup id="cite_ref-E192_14-0" class="reference"><a href="#cite_note-E192-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>To apply this to algebraic curves, write <i>f</i>(<i>x</i>, <i>y</i>) as </p> <dl><dd><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=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f966ef9c5dc747fc2e8b70deb861f55b594c1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.814ex; height:2.509ex;" alt="{\displaystyle f=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,}" /></span></dd></dl> <p>where each <i>u</i><sub><i>r</i></sub> is the sum of all terms of degree <i>r</i>. The homogeneous equation of the curve is then </p> <dl><dd><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=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>z</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>0.</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d132f29baa6539d17c392e0d5b7d5b4baa89bc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.365ex; height:3.009ex;" alt="{\displaystyle g=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,}" /></span></dd></dl> <p>Applying the equation above and setting <i>z</i>=1 produces </p> <dl><dd><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 {\frac {\partial f}{\partial x}}(X,Y)\cdot x+{\frac {\partial f}{\partial y}}(X,Y)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot x+{\frac {\partial f}{\partial y}}(X,Y)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,1)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775ca49bd0f1a1500786d6212bc7d49eeaf4f8f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.958ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot x+{\frac {\partial f}{\partial y}}(X,Y)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,1)=0}" /></span></dd></dl> <p>as the equation of the tangent line.<sup id="cite_ref-E193_15-0" class="reference"><a href="#cite_note-E193-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.<sup id="cite_ref-E192_14-1" class="reference"><a href="#cite_note-E192-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>If the curve is given <a href="/wiki/Parametric_equation" title="Parametric equation">parametrically</a> by </p> <dl><dd><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=x(t),\quad y=y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em"></mspace> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x(t),\quad y=y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ddb9759f2a8af4b9fce3834658f7df87958858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.822ex; height:2.843ex;" alt="{\displaystyle x=x(t),\quad y=y(t)}" /></span></dd></dl> <p>then the slope of the tangent is </p> <dl><dd><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 {\frac {dy}{dx}}={\frac {dy}{dt}}{\bigg /}{\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>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>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="2.047em" minsize="2.047em">/</mo> </mrow> </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}{dx}}={\frac {dy}{dt}}{\bigg /}{\frac {dx}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6083ef2d9a479764e887c418e68720f87fd9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.495ex; height:6.176ex;" alt="{\displaystyle {\frac {dy}{dx}}={\frac {dy}{dt}}{\bigg /}{\frac {dx}{dt}}}" /></span></dd></dl> <p>giving the equation for the tangent line at <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 \,t=T,\,X=x(T),\,Y=y(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace"></mspace> <mi>t</mi> <mo>=</mo> <mi>T</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>Y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,t=T,\,X=x(T),\,Y=y(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0194f4ee71ca5b68aeb6de0abe8b9577b6da58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.13ex; height:2.843ex;" alt="{\displaystyle \,t=T,\,X=x(T),\,Y=y(T)}" /></span> as<sup id="cite_ref-E196_16-0" class="reference"><a href="#cite_note-E196-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\frac {dx}{dt}}(T)\cdot (y-Y)={\frac {dy}{dt}}(T)\cdot (x-X).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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 stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}(T)\cdot (y-Y)={\frac {dy}{dt}}(T)\cdot (x-X).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc2718ad98287cad566b690feaeb5743e79a564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.122ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}(T)\cdot (y-Y)={\frac {dy}{dt}}(T)\cdot (x-X).}" /></span></dd></dl> <p>If </p> <dl><dd><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 {\frac {dx}{dt}}(T)={\frac {dy}{dt}}(T)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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 stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}(T)={\frac {dy}{dt}}(T)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0302a6413b5485016fc4fcb55d025e2650ae5284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.487ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}(T)={\frac {dy}{dt}}(T)=0,}" /></span></dd></dl> <p>the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve. </p> <div class="mw-heading mw-heading3"><h3 id="Normal_line_to_a_curve">Normal line to a curve</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=8" title="Edit section: Normal line to a curve"><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">Further information: <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">Normal (geometry)</a></div> <p>The line perpendicular to the tangent line to a curve at the point of tangency is called the <i>normal line</i> to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is <i>y</i> = <i>f</i>(<i>x</i>) then slope of the normal line is </p> <dl><dd><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{\bigg /}{\frac {dy}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="2.047em" minsize="2.047em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1{\bigg /}{\frac {dy}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132001a249e200119211ae75c789391377e1536b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.778ex; height:6.176ex;" alt="{\displaystyle -1{\bigg /}{\frac {dy}{dx}}}" /></span></dd></dl> <p>and it follows that the equation of the normal line at (X, Y) is </p> <dl><dd><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-X)+{\frac {dy}{dx}}(y-Y)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> <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 stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-X)+{\frac {dy}{dx}}(y-Y)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52cd5adda4f4e23a88e9d85718d7681c3f6a30fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.668ex; height:5.509ex;" alt="{\displaystyle (x-X)+{\frac {dy}{dx}}(y-Y)=0.}" /></span></dd></dl> <p>Similarly, if the equation of the curve has the form <i>f</i>(<i>x</i>, <i>y</i>) = 0 then the equation of the normal line is given by<sup id="cite_ref-E194_17-0" class="reference"><a href="#cite_note-E194-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\frac {\partial f}{\partial y}}(x-X)-{\frac {\partial f}{\partial x}}(y-Y)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial y}}(x-X)-{\frac {\partial f}{\partial x}}(y-Y)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd2311c9b8064e2da8d98e13ad9a6010620cbf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.203ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial f}{\partial y}}(x-X)-{\frac {\partial f}{\partial x}}(y-Y)=0.}" /></span></dd></dl> <p>If the curve is given parametrically by </p> <dl><dd><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=x(t),\quad y=y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em"></mspace> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x(t),\quad y=y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ddb9759f2a8af4b9fce3834658f7df87958858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.822ex; height:2.843ex;" alt="{\displaystyle x=x(t),\quad y=y(t)}" /></span></dd></dl> <p>then the equation of the normal line is<sup id="cite_ref-E196_16-1" class="reference"><a href="#cite_note-E196-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\frac {dx}{dt}}(x-X)+{\frac {dy}{dt}}(y-Y)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <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 stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dx}{dt}}(x-X)+{\frac {dy}{dt}}(y-Y)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/958a8ed2ea23c43ac0a98c14e3ea73cca7184794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.875ex; height:5.509ex;" alt="{\displaystyle {\frac {dx}{dt}}(x-X)+{\frac {dy}{dt}}(y-Y)=0.}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Angle_between_curves">Angle between curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=9" title="Edit section: Angle between curves"><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/Angle#Angles_between_curves" title="Angle">Angle § Angles between curves</a></div> <p>The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.<sup id="cite_ref-E195_18-0" class="reference"><a href="#cite_note-E195-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_tangents_at_a_point">Multiple tangents at a point</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=10" title="Edit section: Multiple tangents at a point"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lima%C3%A7onTrisectrix.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Lima%C3%A7onTrisectrix.svg/300px-Lima%C3%A7onTrisectrix.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Lima%C3%A7onTrisectrix.svg/450px-Lima%C3%A7onTrisectrix.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Lima%C3%A7onTrisectrix.svg/600px-Lima%C3%A7onTrisectrix.svg.png 2x" data-file-width="500" data-file-height="400" /></a><figcaption>The limaçon trisectrix: a curve with two tangents at the origin.</figcaption></figure> <p>The formulas above fail when the point is a <a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">singular point</a>. In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables (or by <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translating</a> the curve) this gives a method for finding the tangent lines at any singular point. </p><p>For example, the equation of the <a href="/wiki/Lima%C3%A7on_trisectrix" title="Limaçon trisectrix">limaçon trisectrix</a> shown to the right is </p> <dl><dd><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^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a325ca3fe826557e7b702f17caa1e8f51b3d5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.53ex; height:3.176ex;" alt="{\displaystyle (x^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,}" /></span></dd></dl> <p>Expanding this and eliminating all but terms of degree 2 gives </p> <dl><dd><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^{2}(3x^{2}-y^{2})=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}(3x^{2}-y^{2})=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6b1fbed355a462704eb1862b11d4654c4595ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.343ex; height:3.176ex;" alt="{\displaystyle a^{2}(3x^{2}-y^{2})=0\,}" /></span></dd></dl> <p>which, when factored, becomes </p> <dl><dd><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=\pm {\sqrt {3}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\pm {\sqrt {3}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd3705c9d3b8a6c3e3e79185012d6afd3c2fb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.137ex; height:2.843ex;" alt="{\displaystyle y=\pm {\sqrt {3}}x.}" /></span></dd></dl> <p>So these are the equations of the two tangent lines through the origin.<sup id="cite_ref-E197_19-0" class="reference"><a href="#cite_note-E197-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because the curve is not differentiable at that point although it is differentiable elsewhere. In this case the <a href="/wiki/Left_and_right_derivative" class="mw-redirect" title="Left and right derivative">left and right derivatives</a> are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point from respectively the left (lower values) or the right (higher values). For example, the curve <i>y</i> = |<i>x</i> | is not differentiable at <i>x</i> = 0: its left and right derivatives have respective slopes −1 and 1; the tangents at that point with those slopes are called the left and right tangents.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve <i>y</i> = <i>x</i> <sup>2/3</sup>, for which both the left and right derivatives at <i>x</i> = 0 are infinite; both the left and right tangent lines have equation <i>x</i> = 0. </p> <div class="mw-heading mw-heading2"><h2 id="Tangent_line_to_a_space_curve">Tangent line to a space curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=11" title="Edit section: Tangent line to a space curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Tangent_vector" title="Tangent vector">Tangent vector</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Tangent_vector&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> is a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> that is tangent to a <a href="/wiki/Curve" title="Curve">curve</a> or <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> at a given point. Tangent vectors are described in the <a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">differential geometry of curves</a> in the context of curves in R<sup><i>n</i></sup>. More generally, tangent vectors are elements of a <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> of a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>. Tangent vectors can also be described in terms of <a href="/wiki/Germ_(mathematics)" title="Germ (mathematics)">germs</a>. Formally, a tangent vector at the point <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> is a linear <a href="/wiki/Derivation_(differential_algebra)" title="Derivation (differential algebra)">derivation</a> of the algebra defined by the set of germs at <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>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Tangent_circles">Tangent circles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=12" title="Edit section: Tangent circles"><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/Tangent_circles" title="Tangent circles">Tangent circles</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Tangent_circles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tangent_circles.svg/200px-Tangent_circles.svg.png" decoding="async" width="200" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tangent_circles.svg/300px-Tangent_circles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tangent_circles.svg/400px-Tangent_circles.svg.png 2x" data-file-width="600" data-file-height="800" /></a><figcaption>Two pairs of tangent circles. Above internally and below externally tangent</figcaption></figure> <p>Two distinct circles lying in the same plane are said to be <i>tangent</i> to each other if they meet at exactly one point. </p><p>If points in the plane are described using <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, then two <a href="/wiki/Circles" class="mw-redirect" title="Circles">circles</a>, with <a href="/wiki/Radii" class="mw-redirect" title="Radii">radii</a> <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 r_{1},r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1},r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00a857c907f8c4b3858f06a1b5c1db4282ab027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.24ex; height:2.009ex;" alt="{\displaystyle r_{1},r_{2}}" /></span> and centers <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_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{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="{\displaystyle (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}" /></span> are tangent to each other whenever </p> <dl><dd><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 \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}\pm r_{2}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>±</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}\pm r_{2}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed64c0d9e58fea16764d4e532ffc23f95494428f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.058ex; height:3.343ex;" alt="{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}\pm r_{2}\right)^{2}.}" /></span></dd></dl> <p>The two circles are called <i>externally tangent</i> if the <a href="/wiki/Distance#Geometry" title="Distance">distance</a> between their centres is equal to the sum of their radii, </p> <dl><dd><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 \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}+r_{2}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}+r_{2}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf27addc760773762e2acd24a7338cbf700c163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.058ex; height:3.343ex;" alt="{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}+r_{2}\right)^{2}.}" /></span></dd></dl> <p>or <i>internally tangent</i> if the distance between their centres is equal to the difference between their radii:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d72b4b34a0f2224a6fe7f26d79cceff77a95224c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.058ex; height:3.343ex;" alt="{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}\right)^{2}.}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Tangent_plane_to_a_surface">Tangent plane to a surface<span class="anchor" id="For_surfaces"></span><span class="anchor" id="Surfaces"></span><span class="anchor" id="Plane"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=13" title="Edit section: Tangent plane to a surface"><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">"Tangent plane" redirects here. For the geographical concept, see <a href="/wiki/Local_tangent_plane" class="mw-redirect" title="Local tangent plane">Local tangent plane</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Differential_geometry_of_surfaces#Tangent_plane" title="Differential geometry of surfaces">Differential geometry of surfaces § Tangent plane</a>, and <a href="/wiki/Parametric_surface#Tangent_plane" title="Parametric surface">Parametric surface § Tangent plane</a></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/Normal_plane_(geometry)" title="Normal plane (geometry)">Normal plane (geometry)</a></div><p>The <b>tangent plane</b> to a <a href="/wiki/Surface_(geometry)" class="mw-redirect" title="Surface (geometry)">surface</a> at a given point <i>p</i> is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at <i>p</i>, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to <i>p</i> as these points converge to <i>p</i>. Mathematically, if the surface is given by 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 z=f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}" /></span>, the equation of the tangent plane at point <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_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}" /></span> can be expressed as: </p><p><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 z-z_{0}={\frac {\partial f}{\partial x}}(x_{0},y_{0})(x-x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y_{0})(y-y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z-z_{0}={\frac {\partial f}{\partial x}}(x_{0},y_{0})(x-x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y_{0})(y-y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819b29bc256fb10a4a5d52722a95c790590b7554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.123ex; height:6.176ex;" alt="{\displaystyle z-z_{0}={\frac {\partial f}{\partial x}}(x_{0},y_{0})(x-x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y_{0})(y-y_{0})}" /></span>. </p><p>Here, <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 {\partial f}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial f}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ea04d099c68b829877f9405b8129b3900382d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.708ex; height:4.176ex;" alt="{\textstyle {\frac {\partial f}{\partial 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="{\textstyle {\frac {\partial f}{\partial y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial f}{\partial y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7dd673d5a4029312a24f82550574b5f14d9de5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:2.672ex; height:4.509ex;" alt="{\textstyle {\frac {\partial f}{\partial y}}}" /></span> are the partial derivatives of 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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> with respect to <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> 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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> respectively, evaluated at the point <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_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" /></span>. In essence, the tangent plane captures the local behavior of the surface at the specific point <i>p</i>. It's a fundamental concept used in calculus and differential geometry, crucial for understanding how functions change locally on surfaces. </p> <div class="mw-heading mw-heading2"><h2 id="Higher-dimensional_manifolds">Higher-dimensional manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=14" title="Edit section: Higher-dimensional manifolds"><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/Tangent_space" title="Tangent space">Tangent space</a></div> <p>More generally, there is a <i>k</i>-dimensional <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> at each point of a <i>k</i>-dimensional <a href="/wiki/Manifold" title="Manifold">manifold</a> in the <i>n</i>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. </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=Tangent&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Multiplicity_(mathematics)#Behavior_of_a_polynomial_function_near_a_multiple_root" title="Multiplicity (mathematics)">Behavior of a polynomial function near a multiple root</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Normal_(geometry)" title="Normal (geometry)">Normal (geometry)</a></li> <li><a href="/wiki/Osculating_circle" title="Osculating circle">Osculating circle</a></li> <li><a href="/wiki/Osculating_curve" title="Osculating curve">Osculating curve</a></li> <li><a href="/wiki/Osculating_plane" title="Osculating plane">Osculating plane</a></li> <li><a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a></li> <li><a href="/wiki/Subtangent" title="Subtangent">Subtangent</a></li> <li><a href="/wiki/Supporting_line" title="Supporting line">Supporting line</a></li> <li><a href="/wiki/Algebraic_curve#Tangent_at_a_point" title="Algebraic curve">Tangent at a point</a></li> <li><a href="/wiki/Tangent_cone" title="Tangent cone">Tangent cone</a></li> <li><a href="/wiki/Tangent_lines_to_circles" title="Tangent lines to circles">Tangent lines to circles</a></li> <li><a href="/wiki/Tangent_vector" title="Tangent vector">Tangent vector</a></li> <li><a href="/wiki/Tangential_angle" title="Tangential angle">Tangential angle</a></li> <li><a href="/wiki/Tangential_and_normal_components" title="Tangential and normal components">Tangential component</a></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=Tangent&action=edit&section=16" 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 mw-references-columns"><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">In "<a href="/wiki/Nova_Methodus_pro_Maximis_et_Minimis" title="Nova Methodus pro Maximis et Minimis">Nova Methodus pro Maximis et Minimis</a>" (<i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i>, Oct. 1684), Leibniz appears to have a notion of tangent lines readily from the start, but later states: "modo teneatur in genere, tangentem invenire esse rectam ducere, quae duo curvae puncta distantiam infinite parvam habentia jungat, seu latus productum polygoni infinitanguli, quod nobis curvae aequivalet", ie. defines the method for drawing tangents through points infinitely close to each other.</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="CITEREFThomas_L._Hankins1985" class="citation book cs1">Thomas L. Hankins (1985). <i>Science and the Enlightenment</i>. Cambridge University Press. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521286190" title="Special:BookSources/9780521286190"><bdi>9780521286190</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+the+Enlightenment&rft.pages=23&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=9780521286190&rft.au=Thomas+L.+Hankins&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Dan Sloughter (2000) . "<a rel="nofollow" class="external text" href="https://math.dartmouth.edu/opencalc2/dcsbook/c3pdf/sec31.pdf">Best Affine Approximations</a>"</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="CITEREFEuclid" class="citation web cs1">Euclid. <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/elements/bookIII/bookIII.html">"Euclid's Elements"</a><span class="reference-accessdate">. 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Retrieved <span class="nowrap">1 June</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=e-CALCULUS+Section+2.8&rft.pages=2.8&rft.aulast=Shenk&rft.aufirst=Al&rft_id=http%3A%2F%2Fmath.ucsd.edu%2F~ashenk%2FSection2_8.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></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="CITEREFKatz2008" class="citation book cs1">Katz, Victor J. (2008). <i>A History of Mathematics</i> (3rd ed.). Addison Wesley. p. 510. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0321387004" title="Special:BookSources/978-0321387004"><bdi>978-0321387004</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=510&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2008&rft.isbn=978-0321387004&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWolfson2001" class="citation journal cs1">Wolfson, Paul R. (2001). "The Crooked Made Straight: Roberval and Newton on Tangents". <i>The American Mathematical Monthly</i>. <b>108</b> (3): <span class="nowrap">206–</span>216. <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%2F2695381">10.2307/2695381</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/2695381">2695381</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+Crooked+Made+Straight%3A+Roberval+and+Newton+on+Tangents&rft.volume=108&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E206-%3C%2Fspan%3E216&rft.date=2001&rft_id=info%3Adoi%2F10.2307%2F2695381&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2695381%23id-name%3DJSTOR&rft.aulast=Wolfson&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" 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="CITEREFKatz2008" class="citation book cs1">Katz, Victor J. (2008). <i>A History of Mathematics</i> (3rd ed.). Addison Wesley. pp. <span class="nowrap">512–</span>514. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0321387004" title="Special:BookSources/978-0321387004"><bdi>978-0321387004</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E512-%3C%2Fspan%3E514&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2008&rft.isbn=978-0321387004&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Noah Webster, <i>American Dictionary of the English Language</i> (New York: S. Converse, 1828), vol. 2, p. 733, <a rel="nofollow" class="external autonumber" href="https://archive.org/stream/americandictiona02websrich#page/n733/mode/2up">[1]</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDescartes1954" class="citation book cs1">Descartes, René (1954) [1637]. <a rel="nofollow" class="external text" href="https://archive.org/details/geometryofrene00desc/page/95/"><i>The Geometry of René Descartes</i></a>. Translated by <a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">Smith, David Eugene</a>; Latham, Marcia L. Open Court. p. 95.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Ren%C3%A9+Descartes&rft.pages=95&rft.pub=Open+Court&rft.date=1954&rft.aulast=Descartes&rft.aufirst=Ren%C3%A9&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofrene00desc%2Fpage%2F95%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFR._E._Langer1937" class="citation journal cs1">R. E. Langer (October 1937). "Rene Descartes". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>. <b>44</b> (8). Mathematical Association of America: <span class="nowrap">495–</span>512. <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%2F2301226">10.2307/2301226</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/2301226">2301226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Rene+Descartes&rft.volume=44&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E495-%3C%2Fspan%3E512&rft.date=1937-10&rft_id=info%3Adoi%2F10.2307%2F2301226&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2301226%23id-name%3DJSTOR&rft.au=R.+E.+Langer&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span> </li> <li id="cite_note-E191-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-E191_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-E191_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Edwards Art. 191</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", <i><a href="/wiki/Mathematical_Gazette" class="mw-redirect" title="Mathematical Gazette">Mathematical Gazette</a></i>, November 2005, 466–467.</span> </li> <li id="cite_note-E192-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-E192_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-E192_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Edwards Art. 192</span> </li> <li id="cite_note-E193-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-E193_15-0">^</a></b></span> <span class="reference-text">Edwards Art. 193</span> </li> <li id="cite_note-E196-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-E196_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-E196_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Edwards Art. 196</span> </li> <li id="cite_note-E194-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-E194_17-0">^</a></b></span> <span class="reference-text">Edwards Art. 194</span> </li> <li id="cite_note-E195-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-E195_18-0">^</a></b></span> <span class="reference-text">Edwards Art. 195</span> </li> <li id="cite_note-E197-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-E197_19-0">^</a></b></span> <span class="reference-text">Edwards Art. 197</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Thomas, George B. Jr., and Finney, Ross L. (1979), <i>Calculus and Analytic Geometry</i>, Addison Wesley Publ. Co.: p. 140.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://homepage.eircom.net/~phabfys/circles.html">"Circles For Leaving Certificate Honours Mathematics by Thomas O'Sullivan 1997"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Circles+For+Leaving+Certificate+Honours+Mathematics+by+Thomas+O%27Sullivan+1997&rft_id=http%3A%2F%2Fhomepage.eircom.net%2F~phabfys%2Fcircles.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent&action=edit&section=17" title="Edit section: Sources"><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 id="CITEREFJ._Edwards1892" class="citation book cs1">J. Edwards (1892). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.109607"><i>Differential Calculus</i></a>. London: MacMillan and Co. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.109607/page/n161">143</a> ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Calculus&rft.place=London&rft.pages=143+ff&rft.pub=MacMillan+and+Co.&rft.date=1892&rft.au=J.+Edwards&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.109607&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></li></ul> <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=Tangent&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Tangency" class="extiw" title="commons:Category:Tangency">Tangency</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735" /><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikisource-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/40px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/60px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/120px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the 1921 <i><a href="/wiki/Collier%27s_Encyclopedia" title="Collier's Encyclopedia">Collier's Encyclopedia</a></i> article <i><b><a href="https://en.wikisource.org/wiki/Collier%27s_New_Encyclopedia_(1921)/Tangent" class="extiw" title="s:Collier's New Encyclopedia (1921)/Tangent">Tangent</a></b></i>.</div></div> </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=Tangent_line">"Tangent line"</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=Tangent+line&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTangent_line&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Tangent_Line"><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/TangentLine.html">"Tangent Line"</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=Tangent+Line&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTangentLine.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATangent" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tangent.html">Tangent to a circle</a> With interactive animation</li> <li><a rel="nofollow" class="external text" href="http://www.vias.org/simulations/simusoft_difftangent.html">Tangent and first derivative</a> — An interactive simulation</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output 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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_topics" title="Template:Calculus topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus_topics" title="Template talk:Calculus topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus_topics" title="Special:EditPage/Template:Calculus topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Calculus249" style="font-size:114%;margin:0 4em"><a href="/wiki/Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" 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 class="mw-selflink selflink">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 href="/wiki/Chain_rule" title="Chain rule">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 class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox390" style="padding:3px"><table class="nowraplinks hlist navbox-inner" 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