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class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Cuprins</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ascunde</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Început</div> </a> </li> <li id="toc-Disputa_Leibnitz–Newton" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Disputa_Leibnitz–Newton"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Disputa Leibnitz–Newton</span> </div> </a> <ul id="toc-Disputa_Leibnitz–Newton-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivată_și_derivabilitate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivată_și_derivabilitate"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Derivată și derivabilitate</span> </div> </a> <ul id="toc-Derivată_și_derivabilitate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcții_derivabile" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcții_derivabile"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Funcții derivabile</span> </div> </a> <button aria-controls="toc-Funcții_derivabile-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 Funcții derivabile subsection</span> </button> <ul id="toc-Funcții_derivabile-sublist" class="vector-toc-list"> <li id="toc-Relația_dintre_continuitate_și_derivabilitate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relația_dintre_continuitate_și_derivabilitate"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Relația dintre continuitate și derivabilitate</span> </div> </a> <ul id="toc-Relația_dintre_continuitate_și_derivabilitate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Derivarea_funcțiilor_obținute_prin_operații_algebrice_elementare" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivarea_funcțiilor_obținute_prin_operații_algebrice_elementare"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Derivarea funcțiilor obținute prin operații algebrice elementare</span> </div> </a> <button aria-controls="toc-Derivarea_funcțiilor_obținute_prin_operații_algebrice_elementare-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 Derivarea funcțiilor obținute prin operații algebrice elementare subsection</span> </button> <ul id="toc-Derivarea_funcțiilor_obținute_prin_operații_algebrice_elementare-sublist" class="vector-toc-list"> <li id="toc-Derivatele_unor_funcții_elementare" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivatele_unor_funcții_elementare"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Derivatele unor funcții elementare</span> </div> </a> <ul id="toc-Derivatele_unor_funcții_elementare-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notații" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notații"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notații</span> </div> </a> <ul id="toc-Notații-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografie</span> </div> </a> <ul id="toc-Bibliografie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vezi_și" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vezi_și"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Vezi și</span> </div> </a> <ul id="toc-Vezi_și-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Legături_externe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legături_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Legături externe</span> </div> </a> <ul id="toc-Legături_externe-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="Cuprins" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Comută cuprinsul" > <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">Comută cuprinsul</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">Derivată</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="Mergeți la un articol în altă limbă. Disponibil în 91 limbi" > <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-91" 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">91 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Afgeleide" title="Afgeleide – afrikaans" lang="af" hreflang="af" data-title="Afgeleide" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%8D%E1%8B%B5%E1%8B%B5%E1%88%AD" title="ውድድር – amharică" lang="am" hreflang="am" data-title="ውድድር" data-language-autonym="አማርኛ" data-language-local-name="amharică" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Derivada" title="Derivada – aragoneză" lang="an" hreflang="an" data-title="Derivada" data-language-autonym="Aragonés" data-language-local-name="aragoneză" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B4%D8%AA%D9%82_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مشتق (رياضيات) – arabă" lang="ar" hreflang="ar" data-title="مشتق (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabă" 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/Derivada" title="Derivada – asturiană" lang="ast" hreflang="ast" data-title="Derivada" data-language-autonym="Asturianu" data-language-local-name="asturiană" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C3%B6r%C9%99m%C9%99" title="Törəmə – azeră" lang="az" hreflang="az" data-title="Törəmə" data-language-autonym="Azərbaycanca" data-language-local-name="azeră" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AA%D8%A4%D8%B1%D9%87%E2%80%8C%D9%85%D9%87" title="تؤره‌مه – South Azerbaijani" lang="azb" hreflang="azb" data-title="تؤره‌مه" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D0%BD%D1%8B%D2%A3_%D1%81%D1%8B%D2%93%D0%B0%D1%80%D1%8B%D0%BB%D0%BC%D0%B0%D2%BB%D1%8B" title="Функцияның сығарылмаһы – bașkiră" lang="ba" hreflang="ba" data-title="Функцияның сығарылмаһы" data-language-autonym="Башҡортса" data-language-local-name="bașkiră" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D1%8B%D1%82%D0%B2%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%96" title="Вытворная функцыі – belarusă" lang="be" hreflang="be" data-title="Вытворная функцыі" data-language-autonym="Беларуская" data-language-local-name="belarusă" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%92%D1%8B%D1%82%D0%B2%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%96" title="Вытворная функцыі – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Вытворная функцыі" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0" title="Производна – bulgară" lang="bg" hreflang="bg" data-title="Производна" data-language-autonym="Български" data-language-local-name="bulgară" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2%E0%A4%A8" title="अवकलन – Bhojpuri" lang="bh" hreflang="bh" data-title="अवकलन" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A6%B0%E0%A6%9C" title="অন্তরজ – bengaleză" lang="bn" hreflang="bn" data-title="অন্তরজ" data-language-autonym="বাংলা" data-language-local-name="bengaleză" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Izvod" title="Izvod – bosniacă" lang="bs" hreflang="bs" data-title="Izvod" data-language-autonym="Bosanski" data-language-local-name="bosniacă" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="articol de calitate"><a href="https://ca.wikipedia.org/wiki/Derivada" title="Derivada – catalană" lang="ca" hreflang="ca" data-title="Derivada" data-language-autonym="Català" data-language-local-name="catalană" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%D8%B1%D8%AA%DB%95" title="گرتە – kurdă centrală" lang="ckb" hreflang="ckb" data-title="گرتە" data-language-autonym="کوردی" data-language-local-name="kurdă centrală" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Derivace" title="Derivace – cehă" lang="cs" hreflang="cs" data-title="Derivace" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BD_%D1%82%C4%83%D1%85%C4%83%D0%BC%C4%95" title="Функцин тăхăмĕ – ciuvașă" lang="cv" hreflang="cv" data-title="Функцин тăхăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Deilliant" title="Deilliant – galeză" lang="cy" hreflang="cy" data-title="Deilliant" data-language-autonym="Cymraeg" data-language-local-name="galeză" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Ableitung_(Mathematik)" title="Ableitung (Mathematik) – germană" lang="de" hreflang="de" data-title="Ableitung (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%AC%CE%B3%CF%89%CE%B3%CE%BF%CF%82" title="Παράγωγος – greacă" lang="el" hreflang="el" data-title="Παράγωγος" data-language-autonym="Ελληνικά" data-language-local-name="greacă" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="articol bun"><a href="https://en.wikipedia.org/wiki/Derivative" title="Derivative – engleză" lang="en" hreflang="en" data-title="Derivative" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Deriva%C4%B5o_(matematiko)" title="Derivaĵo (matematiko) – esperanto" lang="eo" hreflang="eo" data-title="Derivaĵo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Derivada" title="Derivada – spaniolă" lang="es" hreflang="es" data-title="Derivada" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tuletis_(matemaatika)" title="Tuletis (matemaatika) – estonă" lang="et" hreflang="et" data-title="Tuletis (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estonă" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Deribatu" title="Deribatu – bască" lang="eu" hreflang="eu" data-title="Deribatu" data-language-autonym="Euskara" data-language-local-name="bască" 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%D8%B4%D8%AA%D9%82" title="مشتق – persană" lang="fa" hreflang="fa" data-title="مشتق" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Derivaatta" title="Derivaatta – finlandeză" lang="fi" hreflang="fi" data-title="Derivaatta" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9riv%C3%A9e" title="Dérivée – franceză" lang="fr" hreflang="fr" data-title="Dérivée" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Derivade" title="Derivade – friulană" lang="fur" hreflang="fur" data-title="Derivade" data-language-autonym="Furlan" data-language-local-name="friulană" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%ADorthach" title="Díorthach – irlandeză" lang="ga" hreflang="ga" data-title="Díorthach" data-language-autonym="Gaeilge" data-language-local-name="irlandeză" 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/Derivada" title="Derivada – galiciană" lang="gl" hreflang="gl" data-title="Derivada" data-language-autonym="Galego" data-language-local-name="galiciană" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת – ebraică" lang="he" hreflang="he" data-title="נגזרת" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2%E0%A4%9C" title="अवकलज – hindi" lang="hi" hreflang="hi" data-title="अवकलज" 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/Derivacija" title="Derivacija – croată" lang="hr" hreflang="hr" data-title="Derivacija" data-language-autonym="Hrvatski" data-language-local-name="croată" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Deriv%C3%A1lt" title="Derivált – maghiară" lang="hu" hreflang="hu" data-title="Derivált" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%AE%D5%A1%D5%B6%D6%81%D5%B5%D5%A1%D5%AC" title="Ածանցյալ – armeană" lang="hy" hreflang="hy" data-title="Ածանցյալ" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Turunan" title="Turunan – indoneziană" lang="id" hreflang="id" data-title="Turunan" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Derivajo" title="Derivajo – ido" lang="io" hreflang="io" data-title="Derivajo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Aflei%C3%B0a_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Afleiða (stærðfræði) – islandeză" lang="is" hreflang="is" data-title="Afleiða (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="islandeză" 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/Derivata" title="Derivata – italiană" lang="it" hreflang="it" data-title="Derivata" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86" title="微分 – japoneză" lang="ja" hreflang="ja" data-title="微分" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AC%E1%83%90%E1%83%A0%E1%83%9B%E1%83%9D%E1%83%94%E1%83%91%E1%83%A3%E1%83%9A%E1%83%98" title="წარმოებული – georgiană" lang="ka" hreflang="ka" data-title="წარმოებული" data-language-autonym="ქართული" data-language-local-name="georgiană" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Differencial" title="Differencial – karakalpak" lang="kaa" hreflang="kaa" data-title="Differencial" data-language-autonym="Qaraqalpaqsha" data-language-local-name="karakalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AF%B8%EB%B6%84" title="미분 – coreeană" lang="ko" hreflang="ko" data-title="미분" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Derivativum" title="Derivativum – latină" lang="la" hreflang="la" data-title="Derivativum" data-language-autonym="Latina" data-language-local-name="latină" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo badge-Q17437796 badge-featuredarticle mw-list-item" title="articol de calitate"><a href="https://lmo.wikipedia.org/wiki/Derivada" title="Derivada – Lombard" lang="lmo" hreflang="lmo" data-title="Derivada" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%9C%E0%BA%BB%E0%BA%99%E0%BA%95%E0%BA%B3%E0%BA%A5%E0%BA%B2" title="ຜົນຕຳລາ – laoțiană" lang="lo" hreflang="lo" data-title="ຜົນຕຳລາ" data-language-autonym="ລາວ" data-language-local-name="laoțiană" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/I%C5%A1vestin%C4%97" title="Išvestinė – lituaniană" lang="lt" hreflang="lt" data-title="Išvestinė" data-language-autonym="Lietuvių" data-language-local-name="lituaniană" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Atvasin%C4%81jums" title="Atvasinājums – letonă" lang="lv" hreflang="lv" data-title="Atvasinājums" data-language-autonym="Latviešu" data-language-local-name="letonă" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%B7%D0%B2%D0%BE%D0%B4" title="Извод – macedoneană" lang="mk" hreflang="mk" data-title="Извод" data-language-autonym="Македонски" data-language-local-name="macedoneană" 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%85%E0%B4%B5%E0%B4%95%E0%B4%B2%E0%B4%9C%E0%B4%82" title="അവകലജം – malayalam" lang="ml" hreflang="ml" data-title="അവകലജം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2%E0%A4%A8" title="अवकलन – marathi" lang="mr" hreflang="mr" data-title="अवकलन" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Terbitan" title="Terbitan – malaeză" lang="ms" hreflang="ms" data-title="Terbitan" data-language-autonym="Bahasa Melayu" data-language-local-name="malaeză" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Derivata" title="Derivata – malteză" lang="mt" hreflang="mt" data-title="Derivata" data-language-autonym="Malti" data-language-local-name="malteză" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9C%E1%80%AD%E1%80%AF%E1%80%80%E1%80%BA%E1%80%95%E1%80%BC%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%BE%E1%80%AF%E1%80%94%E1%80%BA%E1%80%B8_%E1%80%90%E1%80%BD%E1%80%80%E1%80%BA%E1%80%91%E1%80%AF%E1%80%90%E1%80%BA%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8" title="အလိုက်ပြောင်းနှုန်း တွက်ထုတ်ခြင်း – birmană" lang="my" hreflang="my" data-title="အလိုက်ပြောင်းနှုန်း တွက်ထုတ်ခြင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmană" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Afgeleide" title="Afgeleide – neerlandeză" lang="nl" hreflang="nl" data-title="Afgeleide" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Derivasjon" title="Derivasjon – norvegiană nynorsk" lang="nn" hreflang="nn" data-title="Derivasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegiană nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Derivasjon" title="Derivasjon – norvegiană bokmål" lang="nb" hreflang="nb" data-title="Derivasjon" data-language-autonym="Norsk bokmål" data-language-local-name="norvegiană bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Derivada" title="Derivada – occitană" lang="oc" hreflang="oc" data-title="Derivada" data-language-autonym="Occitan" data-language-local-name="occitană" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Babbaafamaa" title="Babbaafamaa – oromo" lang="om" hreflang="om" data-title="Babbaafamaa" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pochodna_funkcji" title="Pochodna funkcji – poloneză" lang="pl" hreflang="pl" data-title="Pochodna funkcji" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%B4%D8%AA%D9%82" title="مشتق – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مشتق" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Derivada" title="Derivada – portugheză" lang="pt" hreflang="pt" data-title="Derivada" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Производная функции – rusă" lang="ru" hreflang="ru" data-title="Производная функции" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Dirivata_(matim%C3%A0tica)" title="Dirivata (matimàtica) – siciliană" lang="scn" hreflang="scn" data-title="Dirivata (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="siciliană" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Derivative" title="Derivative – scots" lang="sco" hreflang="sco" data-title="Derivative" data-language-autonym="Scots" data-language-local-name="scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Izvod" title="Izvod – sârbo-croată" lang="sh" hreflang="sh" data-title="Izvod" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="sârbo-croată" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Derivative_(mathematics)" title="Derivative (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Derivative (mathematics)" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Deriv%C3%A1cia_(funkcia)" title="Derivácia (funkcia) – slovacă" lang="sk" hreflang="sk" data-title="Derivácia (funkcia)" data-language-autonym="Slovenčina" data-language-local-name="slovacă" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Odvod" title="Odvod – slovenă" lang="sl" hreflang="sl" data-title="Odvod" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Derivati" title="Derivati – albaneză" lang="sq" hreflang="sq" data-title="Derivati" data-language-autonym="Shqip" data-language-local-name="albaneză" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%B7%D0%B2%D0%BE%D0%B4" title="Извод – sârbă" lang="sr" hreflang="sr" data-title="Извод" data-language-autonym="Српски / srpski" data-language-local-name="sârbă" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Derivata" title="Derivata – suedeză" lang="sv" hreflang="sv" data-title="Derivata" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Pochodno" title="Pochodno – Silesian" lang="szl" hreflang="szl" data-title="Pochodno" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%95%E0%AF%88%E0%AE%AF%E0%AE%BF%E0%AE%9F%E0%AE%B2%E0%AF%8D" title="வகையிடல் – tamilă" lang="ta" hreflang="ta" data-title="வகையிடல்" data-language-autonym="தமிழ்" data-language-local-name="tamilă" 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%B8%AD%E0%B8%99%E0%B8%B8%E0%B8%9E%E0%B8%B1%E0%B8%99%E0%B8%98%E0%B9%8C" title="อนุพันธ์ – thailandeză" lang="th" hreflang="th" data-title="อนุพันธ์" data-language-autonym="ไทย" data-language-local-name="thailandeză" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Deribatibo" title="Deribatibo – tagalog" lang="tl" hreflang="tl" data-title="Deribatibo" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/T%C3%BCrev" title="Türev – turcă" lang="tr" hreflang="tr" data-title="Türev" data-language-autonym="Türkçe" data-language-local-name="turcă" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%87%D1%8B%D0%B3%D0%B0%D1%80%D1%8B%D0%BB%D0%BC%D0%B0%D1%81%D1%8B" title="Функция чыгарылмасы – tătară" lang="tt" hreflang="tt" data-title="Функция чыгарылмасы" data-language-autonym="Татарча / tatarça" data-language-local-name="tătară" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%85%D1%96%D0%B4%D0%BD%D0%B0" title="Похідна – ucraineană" lang="uk" hreflang="uk" data-title="Похідна" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B4%D8%AA%D9%82" title="مشتق – urdu" lang="ur" hreflang="ur" data-title="مشتق" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Differensial" title="Differensial – uzbecă" lang="uz" hreflang="uz" data-title="Differensial" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbecă" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Derivada" title="Derivada – venetă" lang="vec" hreflang="vec" data-title="Derivada" data-language-autonym="Vèneto" data-language-local-name="venetă" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1o_h%C3%A0m" title="Đạo hàm – vietnameză" lang="vi" hreflang="vi" data-title="Đạo hàm" data-language-autonym="Tiếng Việt" data-language-local-name="vietnameză" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Deriv%C3%AAye" title="Derivêye – valonă" lang="wa" hreflang="wa" data-title="Derivêye" data-language-autonym="Walon" data-language-local-name="valonă" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B0%8E%E6%95%B8" title="導數 – chineză wu" lang="wuu" hreflang="wuu" data-title="導數" data-language-autonym="吴语" data-language-local-name="chineză wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%93%D7%A2%D7%A8%D7%99%D7%95%D7%95%D7%90%D7%98%D7%99%D7%95%D7%95" title="דעריוואטיוו – idiș" lang="yi" hreflang="yi" data-title="דעריוואטיוו" data-language-autonym="ייִדיש" data-language-local-name="idiș" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="articol bun"><a href="https://zh.wikipedia.org/wiki/%E5%AF%BC%E6%95%B0" title="导数 – chineză" lang="zh" hreflang="zh" data-title="导数" data-language-autonym="中文" data-language-local-name="chineză" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link 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typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier: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/320px-Graph_of_sliding_derivative_line.gif" decoding="async" width="320" height="320" 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> În fiecare punct, derivata funcției <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 \scriptstyle f(x)=1+x\sin x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f(x)=1+x\sin x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a84817dd4527cc0edfdaaceeb89a1c8ee914eed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.008ex; height:2.343ex;" alt="{\displaystyle \scriptstyle f(x)=1+x\sin x^{2}}"></span> este <a href="/w/index.php?title=Pant%C4%83_(matematic%C4%83)&action=edit&redlink=1" class="new" title="Pantă (matematică) — pagină inexistentă">panta</a> (înclinarea) <a href="/wiki/Dreapt%C4%83" title="Dreaptă">dreptei</a> care este <a href="/wiki/Tangent%C4%83_(geometrie)" title="Tangentă (geometrie)">tangentă</a> la <a href="/wiki/Curb%C4%83" title="Curbă">curbă</a>. Dreapta care se mișcă este tangenta instantanee la <a href="/wiki/Categorie:Curbe_plane" title="Categorie:Curbe plane">curbă</a> în orice moment; este colorată în <a href="/wiki/Verde" title="Verde">verde</a> dacă este pozitivă, în <a href="/wiki/Negru" title="Negru">negru</a> dacă este <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a>, respectiv în <a href="/wiki/Ro%C8%99u" title="Roșu">roșu</a>, dacă este negativă.</figcaption></figure> <p>În <a href="/wiki/Matematic%C4%83" title="Matematică">matematică</a>, <b>derivata</b> unei <a href="/wiki/Func%C8%9Bie" title="Funcție">funcții</a> este unul dintre conceptele fundamentale ale <a href="/wiki/Analiz%C4%83_matematic%C4%83" class="mw-redirect" title="Analiză matematică">analizei matematice</a>, împreună cu <a href="/wiki/Primitiv%C4%83" title="Primitivă">primitiva</a> (inversa derivatei, adică integrala). </p><p>Derivata unei funcții într-un <a href="/wiki/Punct" class="mw-redirect mw-disambig" title="Punct">punct</a> semnifică rata cu care se modifică <a href="/wiki/Valoare_(matematic%C4%83)" title="Valoare (matematică)">valoarea</a> funcției atunci când se modifică <a href="/wiki/Argumentul_unei_func%C8%9Bii" title="Argumentul unei funcții">argumentul</a>. Cu alte cuvinte, derivata este o formulare matematică a noțiunii de <b>rată de variație</b>. Derivata este un concept foarte versatil, care poate fi privit în multe feluri. De exemplu, referindu-ne la <a href="/wiki/Graficul_unei_func%C8%9Bii" title="Graficul unei funcții">graficul</a> bidimensional al funcției <i>f</i>, derivata într-un punct <i>x</i> reprezintă <a href="/wiki/Pant%C4%83" title="Pantă">panta</a> <a href="/wiki/Tangent%C4%83_(geometrie)" title="Tangentă (geometrie)">tangentei</a> la grafic în punctul <i>x</i>. Panta tangentei se poate aproxima printr-o <a href="/wiki/Secant%C4%83" title="Secantă">secantă</a>. Cu această interpretare geometrică, nu este surprinzător faptul că <a href="/wiki/Examinarea_derivatelor" title="Examinarea derivatelor">derivatele pot fi folosite</a> pentru a descrie multe proprietăți geometrice ale graficelor de funcții, cum ar fi <a href="/wiki/Concavitate" title="Concavitate">concavitatea</a> și <a href="/wiki/Convexitate" title="Convexitate">convexitatea</a>. </p><p>Trebuie menționat că nu toate funcțiile admit derivate. De exemplu, funcțiile nu au derivate în punctele în care au o tangentă verticală, în punctele de <a href="/wiki/Continuitate_(matematic%C4%83)" class="mw-redirect" title="Continuitate (matematică)">discontinuitate</a> și în <a href="/wiki/Punct_de_%C3%AEntoarcere" title="Punct de întoarcere">punctele de întoarcere</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Disputa_Leibnitz–Newton"><span id="Disputa_Leibnitz.E2.80.93Newton"></span>Disputa Leibnitz–Newton</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=1" title="Modifică secțiunea: Disputa Leibnitz–Newton" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=1" title="Edit section's source code: Disputa Leibnitz–Newton"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Calcul_infinitezimal" title="Calcul infinitezimal">Calculul diferențial și cel integral</a> au fost inventate practic simultan, dar independent unul de celălalt, de către englezul <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> (1643–1727), respectiv de către matematicianul german <a href="/wiki/Gottfried_Wilhelm_von_Leibniz" title="Gottfried Wilhelm von Leibniz">Gottfried Wilhelm von Leibniz</a> (1646–1716). </p><p>Se poate menționa, cu titlul aproape anecdotic, dar absolut real, că lumea științifică a momentului respectiv (<a href="/wiki/1685" title="1685">1685</a>-<a href="/wiki/1690" title="1690">1690</a>) asista, aproape „cu sufletul la gură”, timp de câțiva ani buni, la un dialog deschis și permanent al celor doi titani, Leibnitz și Newton. Doar după ce cei doi oameni de știință au ajuns la înțelegerea abordării conceptelor și noțiunilor din ambele puncte de vedere (al fizicianului și al matematicianului), după ce s-au pus de acord cu noțiunile preliminare, limitele și metodologia de <a href="/wiki/Abordare" class="mw-disambig" title="Abordare">abordare</a> a conceptelor etc., cei doi au putut explica și restului lumii științifice despre ce este vorba. </p> <div class="mw-heading mw-heading2"><h2 id="Derivată_și_derivabilitate"><span id="Derivat.C4.83_.C8.99i_derivabilitate"></span>Derivată și derivabilitate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=2" title="Modifică secțiunea: Derivată și derivabilitate" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=2" title="Edit section's source code: Derivată și derivabilitate"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Derivata</b> a apărut din necesitatea de a exprima rata cu care se modifică (variază) o cantitate <i>y</i> ca urmare a modificării (variației) unei alte cantități <i>x</i> de care este legată printr-o funcție. Folosind simbolul Δ pentru a nota modificarea (variația) unei cantități, această rată se definește ca <a href="/wiki/Limit%C4%83_(matematic%C4%83)" title="Limită (matematică)">limita</a> raportului variațiilor (diferențelor): </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 {\Delta y}{\Delta x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta y}{\Delta x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5100256c84fd871304993b43abf54aa86c167aaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.102ex; height:5.676ex;" alt="{\displaystyle {\frac {\Delta y}{\Delta x}}}"></span></dd></dl> <p>pe măsură ce Δ <i>x</i> tinde spre 0 sau altfel exprimat Δ <i>x</i> e în <a href="/wiki/Vecin%C4%83tate_(matematic%C4%83)" title="Vecinătate (matematică)">vecinătatea</a> lui 0. În notația lui <a href="/wiki/Gottfried_Wilhelm_von_Leibniz" title="Gottfried Wilhelm von Leibniz">Leibniz</a>, derivata lui <i>y</i> în raport cu <i>x</i> se scrie </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}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ceb16b58a91d26cf1e442d0682dfa7a2c0ab72c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.382ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dx}}}"></span></dd></dl> <p>sugerând <a href="/wiki/Diferen%C8%9Be_divizate" title="Diferențe divizate">raportul a două diferențe numerice</a> (cantități) <a href="/wiki/Infinitezimal" title="Infinitezimal">infinitezimale</a> (în <a href="/wiki/Vecin%C4%83tate_(matematic%C4%83)" title="Vecinătate (matematică)">vecinătatea</a> lui 0). Expresia de mai sus se poate pronunța fie "<i>dy supra dx</i>", fie "<i>dy la dx</i>". </p><p>În limbajul matematic contemporan, nu se mai face referire la cantitățile care variază; derivata este considerată o operație matematică asupra funcțiilor. Definiția formală a acestei operații (care nu mai face uz de noțiunea de cantități <i><a href="/wiki/Infinitezimal" title="Infinitezimal">infinitezimale</a></i>) este dată de limita când <i>h</i> tinde la 0 (e în <a href="/wiki/Vecin%C4%83tate_(matematic%C4%83)" title="Vecinătate (matematică)">vecinătatea</a> lui 0) a următoarei expresii: </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 {f(x+h)-f(x)}{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>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(x+h)-f(x)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5786e28a8f6398c48edcf26eff7d3a154c1c7fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.338ex; height:5.843ex;" alt="{\displaystyle {\frac {f(x+h)-f(x)}{h}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Funcții_derivabile"><span id="Func.C8.9Bii_derivabile"></span>Funcții derivabile</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=3" title="Modifică secțiunea: Funcții derivabile" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=3" title="Edit section's source code: Funcții derivabile"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O funcție <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> este <a href="/wiki/Func%C8%9Bie_derivabil%C4%83" title="Funcție derivabilă">derivabilă</a> într-un punct <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> dacă: </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 \exists \lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6395d1818beb5d3d6bfb387834a1f87e522bdf6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.669ex; height:6.009ex;" alt="{\displaystyle \exists \lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}=L}"></span> și <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 L\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0913a2035a5674d95af33a6c3fb41d2e146dc273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.102ex; height:2.176ex;" alt="{\displaystyle L\in \mathbb {R} }"></span></dd></dl> <p>Dacă <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 L\in \lbrace -\infty ,+\infty \rbrace }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\in \lbrace -\infty ,+\infty \rbrace }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10da08624d3406cb6a960015f297b9c66c9ea654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.046ex; height:2.843ex;" alt="{\displaystyle L\in \lbrace -\infty ,+\infty \rbrace }"></span> atunci spunem că <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> are derivată dar nu este derivabilă </p> <div class="mw-heading mw-heading3"><h3 id="Relația_dintre_continuitate_și_derivabilitate"><span id="Rela.C8.9Bia_dintre_continuitate_.C8.99i_derivabilitate"></span>Relația dintre continuitate și derivabilitate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=4" title="Modifică secțiunea: Relația dintre continuitate și derivabilitate" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=4" title="Edit section's source code: Relația dintre continuitate și derivabilitate"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fie <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:D\subseteq \mathbb {R} \rightarrow \mathbb {R} ,x_{0}\in D\cap D\prime }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>D</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>D</mi> <mo>∩</mo> <mi>D</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:D\subseteq \mathbb {R} \rightarrow \mathbb {R} ,x_{0}\in D\cap D\prime }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f0a3b47528a1ccc2fe0a058818f70bc544476f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.538ex; height:2.509ex;" alt="{\displaystyle f:D\subseteq \mathbb {R} \rightarrow \mathbb {R} ,x_{0}\in D\cap D\prime }"></span> unde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\prime }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi class="MJX-variant" mathvariant="normal">′</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\prime }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d70151c78794f63bd9b485874a21810ca9cb2aed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.564ex; height:2.176ex;" alt="{\displaystyle D\prime }"></span> este mulțimea <a href="/wiki/Punct_de_acumulare_(matematic%C4%83)" title="Punct de acumulare (matematică)">punctelor de acumulare</a>. Atunci: </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}"> <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> derivabilă în <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}\Rightarrow f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">⇒</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\Rightarrow f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6bbf1d6a6eb6575f87c3e28a6cc20b79dd8e658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.277ex; height:2.509ex;" alt="{\displaystyle x_{0}\Rightarrow f}"></span> continuă în <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>, dar :<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> poate fi <a href="/wiki/Func%C8%9Bie_continu%C4%83" title="Funcție continuă">continuă</a> și nederivabilă (conversa afirmației este falsă)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Derivarea_funcțiilor_obținute_prin_operații_algebrice_elementare"><span id="Derivarea_func.C8.9Biilor_ob.C8.9Binute_prin_opera.C8.9Bii_algebrice_elementare"></span>Derivarea funcțiilor obținute prin operații algebrice elementare</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=5" title="Modifică secțiunea: Derivarea funcțiilor obținute prin operații algebrice elementare" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=5" title="Edit section's source code: Derivarea funcțiilor obținute prin operații algebrice elementare"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g:\mathbb {R} \rightarrow \mathbb {R} ,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g:\mathbb {R} \rightarrow \mathbb {R} ,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbe152a2e163242929140e9493df6c23f6aa583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.648ex; height:2.509ex;" alt="{\displaystyle f,g:\mathbb {R} \rightarrow \mathbb {R} ,f}"></span> și <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> funcții derivabile pe domeniul lor de definiție. Atunci: </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'+g'=(f+g)'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'+g'=(f+g)'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245bd3461a48d9be5dd3761988a0c35f8dd7803d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.476ex; height:3.009ex;" alt="{\displaystyle f'+g'=(f+g)'}"></span> ;</dd> <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 (\lambda f)'=\lambda f'\ ,\,\lambda \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mi>λ</mi> <msup> <mi>f</mi> <mo>′</mo> </msup> <mtext> </mtext> <mo>,</mo> <mspace width="thinmathspace" /> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda f)'=\lambda f'\ ,\,\lambda \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/035bab8ee0c08b1a985e7a4cf77091165d150c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.462ex; height:3.009ex;" alt="{\displaystyle (\lambda f)'=\lambda f'\ ,\,\lambda \in \mathbb {R} }"></span> ;</dd> <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 (fg)'=f'g+fg'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mi>g</mi> <mo>+</mo> <mi>f</mi> <msup> <mi>g</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (fg)'=f'g+fg'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdcc2675f11f31c41f08005f4428657ede1b7c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.03ex; height:3.009ex;" alt="{\displaystyle (fg)'=f'g+fg'}"></span> ;</dd> <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 {\biggl (}{\frac {f}{g}}{\biggr )}'={\frac {f'g-fg'}{g^{2}}},\quad g(x)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mi>g</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mi>g</mi> <mo>−</mo> <mi>f</mi> <msup> <mi>g</mi> <mo>′</mo> </msup> </mrow> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl (}{\frac {f}{g}}{\biggr )}'={\frac {f'g-fg'}{g^{2}}},\quad g(x)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539b61c23428a468f4a2bc1a2c665e44cdca7c03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.071ex; height:6.343ex;" alt="{\displaystyle {\biggl (}{\frac {f}{g}}{\biggr )}'={\frac {f'g-fg'}{g^{2}}},\quad g(x)\neq 0}"></span>.</dd> <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\circ f)(x))'=(g(f(x))'=g'(f(x))\cdot f'(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((g\circ f)(x))'=(g(f(x))'=g'(f(x))\cdot f'(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e7dacb96f00666e3cb633e5839ced46bf1ee4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.014ex; height:3.009ex;" alt="{\displaystyle ((g\circ f)(x))'=(g(f(x))'=g'(f(x))\cdot f'(x)}"></span></dd></dl> <p>Aceste <a href="/wiki/Egalitate_(matematic%C4%83)" title="Egalitate (matematică)">egalități</a> se pot demonstra pornind de la definiția derivatei. </p> <div class="mw-heading mw-heading3"><h3 id="Derivatele_unor_funcții_elementare"><span id="Derivatele_unor_func.C8.9Bii_elementare"></span>Derivatele unor funcții elementare</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=6" title="Modifică secțiunea: Derivatele unor funcții elementare" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=6" title="Edit section's source code: Derivatele unor funcții elementare"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Putere:</li></ul> <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(x)=x^{r}}"> <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> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b12c5ad6dc7d9310aef2097252a8c704e58612f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.819ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{r}}"></span></dd></dl> <p>unde <i>r</i> este număr <a href="/wiki/Num%C4%83r_real" title="Număr real">real</a>, atunci: </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'(x)=rx^{r-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)=rx^{r-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbfd6111d4ee4c88930a3eb2f5378decb7c429a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.695ex; height:3.176ex;" alt="{\displaystyle f'(x)=rx^{r-1}}"></span></dd></dl> <p>oriunde derivata este <a href="/wiki/Expresie_bine_definit%C4%83" title="Expresie bine definită">bine definită</a>. </p> <ul><li><i>Funcția <a href="/wiki/Func%C8%9Bie_exponen%C8%9Bial%C4%83" title="Funcție exponențială">exponențială</a> și <a href="/wiki/Logaritm" title="Logaritm">logaritmică</a></i>:</li></ul> <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 {d \over dx}a^{x}=a^{x}ln(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}a^{x}=a^{x}ln(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250fef771c195f54c5e16262091fee91b66f3d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.412ex; height:5.509ex;" alt="{\displaystyle {d \over dx}a^{x}=a^{x}ln(a)}"></span></dd> <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 {d \over dx}log_{a}x={1 \over x\ ln(a)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>l</mi> <mi>o</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mtext> </mtext> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}log_{a}x={1 \over x\ ln(a)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b4cd235703051fcd8048469384c2c48f9cd03e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.715ex; height:6.176ex;" alt="{\displaystyle {d \over dx}log_{a}x={1 \over x\ ln(a)}}"></span></dd></dl> <ul><li><i><a href="/wiki/Func%C8%9Bie_trigonometric%C4%83" title="Funcție trigonometrică">Funcții trigonometrice:</a></i></li></ul> <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 {d \over dx}sin(x)=cos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}sin(x)=cos(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aff0a90aa521094f0efaaba6e340ef42a719e69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.271ex; height:5.509ex;" alt="{\displaystyle {d \over dx}sin(x)=cos(x)}"></span></dd> <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 {d \over dx}cos(x)=-sin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}cos(x)=-sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfd3963ab3ca9e9fb13d4defe766cfde8ff45d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.079ex; height:5.509ex;" alt="{\displaystyle {d \over dx}cos(x)=-sin(x)}"></span></dd> <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 {d \over dx}tan(x)={1 \over cos^{2}(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}tan(x)={1 \over cos^{2}(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae4c6f5b4d3ebeb8d9fbaaf8e2321044c4492c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.338ex; height:6.176ex;" alt="{\displaystyle {d \over dx}tan(x)={1 \over cos^{2}(x)}}"></span></dd> <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 {d \over dx}cot(x)=-{1 \over sin^{2}(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>c</mi> <mi>o</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mi>i</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}cot(x)=-{1 \over sin^{2}(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1501ae46408700f8052526d880c7f0129e24ff7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.718ex; height:6.176ex;" alt="{\displaystyle {d \over dx}cot(x)=-{1 \over sin^{2}(x)}}"></span></dd> <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 {d \over dx}sinh(x)=cosh(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}sinh(x)=cosh(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01363f30979042b841e81d0ee033c59b2785bf01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.949ex; height:5.509ex;" alt="{\displaystyle {d \over dx}sinh(x)=cosh(x)}"></span></dd> <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 {d \over dx}cosh(x)=sinh(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}cosh(x)=sinh(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d31d651a04bef319eab0a29abd35da6d7570ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.949ex; height:5.509ex;" alt="{\displaystyle {d \over dx}cosh(x)=sinh(x)}"></span></dd> <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 {d \over dx}tanh(x)=sech^{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>e</mi> <mi>c</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}tanh(x)=sech^{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c303f1f7034231c7e1379850a834e27bfc31215e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.135ex; height:5.509ex;" alt="{\displaystyle {d \over dx}tanh(x)=sech^{2}(x)}"></span></dd></dl> <ul><li><i>Funcții trigonometrice inverse</i>:</li></ul> <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 {d \over dx}arcsin(x)={1 \over {\sqrt {1-x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arcsin(x)={1 \over {\sqrt {1-x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595a2ffa9e15a0bbc0d1055d402330fdb5187f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.739ex; height:6.676ex;" alt="{\displaystyle {d \over dx}arcsin(x)={1 \over {\sqrt {1-x^{2}}}}}"></span></dd> <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 {d \over dx}arccos(x)=-{1 \over {\sqrt {1-x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arccos(x)=-{1 \over {\sqrt {1-x^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090b440ea68238669e699e3bb991358897aa89e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.484ex; height:6.676ex;" alt="{\displaystyle {d \over dx}arccos(x)=-{1 \over {\sqrt {1-x^{2}}}}}"></span></dd> <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 {d \over dx}arctan(x)={1 \over 1+x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arctan(x)={1 \over 1+x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0fa84921405d7063629597704b28f61cc5aade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.591ex; height:5.843ex;" alt="{\displaystyle {d \over dx}arctan(x)={1 \over 1+x^{2}}}"></span></dd> <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 {d \over dx}arccot(x)=-{1 \over 1+x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>c</mi> <mi>o</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arccot(x)=-{1 \over 1+x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f644db5692b81725579a190569424f95a57749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.91ex; height:5.843ex;" alt="{\displaystyle {d \over dx}arccot(x)=-{1 \over 1+x^{2}}}"></span></dd> <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 {d \over dx}arcsinh(x)={1 \over {\sqrt {x^{2}+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arcsinh(x)={1 \over {\sqrt {x^{2}+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f838021db9ab29515c2f68766d9a979783caec67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.078ex; height:6.676ex;" alt="{\displaystyle {d \over dx}arcsinh(x)={1 \over {\sqrt {x^{2}+1}}}}"></span></dd> <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 {d \over dx}arccosh(x)={1 \over {\sqrt {x^{2}-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arccosh(x)={1 \over {\sqrt {x^{2}-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb84729f747cff600aab7b3361952934ac5defca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.015ex; height:6.676ex;" alt="{\displaystyle {d \over dx}arccosh(x)={1 \over {\sqrt {x^{2}-1}}}}"></span></dd> <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 {d \over dx}arctanh(x)={1 \over 1-x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arctanh(x)={1 \over 1-x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c6727cd81a66841ea422a8467492b764df5f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.93ex; height:5.843ex;" alt="{\displaystyle {d \over dx}arctanh(x)={1 \over 1-x^{2}}}"></span></dd> <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 {d \over dx}arccoth(x)={1 \over 1-x^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>c</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}arccoth(x)={1 \over 1-x^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a963ef9640e714646afa6179c988955eeaaef6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.44ex; height:5.843ex;" alt="{\displaystyle {d \over dx}arccoth(x)={1 \over 1-x^{2}}}"></span></dd> <dd>Valabile pentru domeniile corespunzătoare de definiție.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Notații"><span id="Nota.C8.9Bii"></span>Notații</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=7" title="Modifică secțiunea: Notații" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=7" title="Edit section's source code: Notații"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dacă <i>f</i> este o funcție, derivata funcției <i>f</i> în punctul <i>x</i> se poate nota (simboliza) în mai multe moduri: </p> <ul><li><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)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339dba66f3d167858633a0afbfdcf794302b9eb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.467ex; height:3.009ex;" alt="{\displaystyle f'(x)\quad }"></span></li></ul> <p>pronunțat "<i>f <a href="/w/index.php?title=Prim_(semn)&action=edit&redlink=1" class="new" title="Prim (semn) — pagină inexistentă">prim</a> de x</i>"; </p> <ul><li><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 {d}{dx}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6818ebd87a15d28471e6742720ed7820c79c0a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.799ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}f(x)}"></span></li></ul> <p>pronunțat "<i>d pe d x din f de x</i>"; </p> <ul><li><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 {df}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {df}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d4eb531911adb8362a989a2c6b9e10bd46c099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.382ex; height:5.509ex;" alt="{\displaystyle {\frac {df}{dx}}}"></span></li></ul> <p>pronunțat "<i>d f pe d x</i>" </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{x}f\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>f</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{x}f\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87c21a002f28f099938778c8b058ca5ac7c824f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.698ex; height:2.509ex;" alt="{\displaystyle D_{x}f\quad }"></span></li></ul> <p>pronunțat "<i>d indice x de f</i>". </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografie">Bibliografie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=8" title="Modifică secțiunea: Bibliografie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=8" title="Edit section's source code: Bibliografie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r12727094">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{list-style-type:none;margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>dd{margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin" style=""> <ul><li>Gh. Sirețchi, <i>Analiză matematică</i>, Editura didactică și pedagogică.</li> <li><cite id="CITEREFAntonBivensDavis2005" class="citation">Anton, Howard; Bivens, Irl; Davis, Stephen (<time datetime="2005-02-02">2 februarie 2005</time>), <i>Calculus: Early Transcendentals Single and Multivariable</i> (ed. 8th), New York: Wiley, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-471-47244-5" title="Special:Referințe în cărți/978-0-471-47244-5">978-0-471-47244-5</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendentals+Single+and+Multivariable&rft.place=New+York&rft.edition=8th&rft.pub=Wiley&rft.date=2005-02-02&rft.isbn=978-0-471-47244-5&rft.aulast=Anton&rft.aufirst=Howard&rft.au=Bivens%2C+Irl&rft.au=Davis%2C+Stephen&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><style data-mw-deduplicate="TemplateStyles:r16236537">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"„""”""«""»"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style></li> <li><cite id="CITEREFApostol1967" class="citation"><a href="/w/index.php?title=Tom_M._Apostol&action=edit&redlink=1" class="new" title="Tom M. Apostol — pagină inexistentă">Apostol, Tom M.</a> (iunie 1967), <i>Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra</i>, <b>1</b> (ed. 2nd), Wiley, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-471-00005-1" title="Special:Referințe în cărți/978-0-471-00005-1">978-0-471-00005-1</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%2C+Vol.+1%3A+One-Variable+Calculus+with+an+Introduction+to+Linear+Algebra&rft.edition=2nd&rft.pub=Wiley&rft.date=1967-06&rft.isbn=978-0-471-00005-1&rft.aulast=Apostol&rft.aufirst=Tom+M.&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFApostol1969" class="citation">Apostol, Tom M. (iunie 1969), <i>Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications</i>, <b>1</b> (ed. 2nd), Wiley, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-471-00007-5" title="Special:Referințe în cărți/978-0-471-00007-5">978-0-471-00007-5</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%2C+Vol.+2%3A+Multi-Variable+Calculus+and+Linear+Algebra+with+Applications&rft.edition=2nd&rft.pub=Wiley&rft.date=1969-06&rft.isbn=978-0-471-00007-5&rft.aulast=Apostol&rft.aufirst=Tom+M.&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFCourantJohn1998" class="citation">Courant, Richard; John, Fritz (<time datetime="1998-12-22">22 decembrie 1998</time>), <i>Introduction to Calculus and Analysis, Vol. 1</i>, Springer-Verlag, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-3-540-65058-4" title="Special:Referințe în cărți/978-3-540-65058-4">978-3-540-65058-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Calculus+and+Analysis%2C+Vol.+1&rft.pub=Springer-Verlag&rft.date=1998-12-22&rft.isbn=978-3-540-65058-4&rft.aulast=Courant&rft.aufirst=Richard&rft.au=John%2C+Fritz&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFEves1990" class="citation">Eves, Howard (<time datetime="1990-01-02">2 ianuarie 1990</time>), <i>An Introduction to the History of Mathematics</i> (ed. 6th), Brooks Cole, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-03-029558-4" title="Special:Referințe în cărți/978-0-03-029558-4">978-0-03-029558-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+History+of+Mathematics&rft.edition=6th&rft.pub=Brooks+Cole&rft.date=1990-01-02&rft.isbn=978-0-03-029558-4&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFLarsonHostetlerEdwards2006" class="citation">Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (<time datetime="2006-02-28">28 februarie 2006</time>), <i>Calculus: Early Transcendental Functions</i> (ed. 4th), Houghton Mifflin Company, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-618-60624-5" title="Special:Referințe în cărți/978-0-618-60624-5">978-0-618-60624-5</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Early+Transcendental+Functions&rft.edition=4th&rft.pub=Houghton+Mifflin+Company&rft.date=2006-02-28&rft.isbn=978-0-618-60624-5&rft.aulast=Larson&rft.aufirst=Ron&rft.au=Hostetler%2C+Robert+P.&rft.au=Edwards%2C+Bruce+H.&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFSpivak1994" class="citation"><a href="/w/index.php?title=Michael_Spivak&action=edit&redlink=1" class="new" title="Michael Spivak — pagină inexistentă">Spivak, Michael</a> (septembrie 1994), <i>Calculus</i> (ed. 3rd), Publish or Perish, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-914098-89-8" title="Special:Referințe în cărți/978-0-914098-89-8">978-0-914098-89-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=3rd&rft.pub=Publish+or+Perish&rft.date=1994-09&rft.isbn=978-0-914098-89-8&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFStewart2002" class="citation">Stewart, James (<time datetime="2002-12-24">24 decembrie 2002</time>), <i>Calculus</i> (ed. 5th), Brooks Cole, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-534-39339-7" title="Special:Referințe în cărți/978-0-534-39339-7">978-0-534-39339-7</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.edition=5th&rft.pub=Brooks+Cole&rft.date=2002-12-24&rft.isbn=978-0-534-39339-7&rft.aulast=Stewart&rft.aufirst=James&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFThompson1998" class="citation">Thompson, Silvanus P. (<time datetime="1998-09-08">8 septembrie 1998</time>), <i>Calculus Made Easy</i> (ed. Revised, Updated, Expanded), New York: St. Martin's Press, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-0-312-18548-0" title="Special:Referințe în cărți/978-0-312-18548-0">978-0-312-18548-0</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+Made+Easy&rft.place=New+York&rft.edition=Revised%2C+Updated%2C+Expanded&rft.pub=St.+Martin%27s+Press&rft.date=1998-09-08&rft.isbn=978-0-312-18548-0&rft.aulast=Thompson&rft.aufirst=Silvanus+P.&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Vezi_și"><span id="Vezi_.C8.99i"></span>Vezi și</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=9" title="Modifică secțiunea: Vezi și" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=9" title="Edit section's source code: Vezi și"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tabel_de_derivate" title="Tabel de derivate">Tabel de derivate</a></li> <li><a href="/wiki/Derivat%C4%83_par%C8%9Bial%C4%83" title="Derivată parțială">Derivată parțială</a></li> <li><a href="/w/index.php?title=Derivat%C4%83_func%C8%9Bional%C4%83&action=edit&redlink=1" class="new" title="Derivată funcțională — pagină inexistentă">Derivată funcțională</a></li> <li><a href="/wiki/Primitiv%C4%83" title="Primitivă">Primitivă</a></li> <li><a href="/wiki/Factor_integrant" title="Factor integrant">Factor integrant</a></li> <li><a href="/wiki/Teoria_nodurilor" title="Teoria nodurilor">Teoria nodurilor</a></li> <li><a href="/wiki/Spa%C8%9Biu_topologic" title="Spațiu topologic">Spațiu topologic</a></li> <li><a href="/wiki/Ecua%C8%9Bie_diferen%C8%9Bial%C4%83_ordinar%C4%83" title="Ecuație diferențială ordinară">Ecuație diferențială ordinară</a></li> <li><a href="/wiki/Func%C8%9Bie_continu%C4%83" title="Funcție continuă">Funcție continuă</a></li> <li><a href="/wiki/Limit%C4%83_a_unei_func%C8%9Bii" title="Limită a unei funcții">Limită a unei funcții</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Derivat%C4%83&veaction=edit&section=10" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Derivat%C4%83&action=edit&section=10" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFCrowell2003" class="citation">Crowell, Benjamin (<time datetime="2003">2003</time>), <a rel="nofollow" class="external text" href="http://www.lightandmatter.com/calc/"><i>Calculus</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.date=2003&rft.aulast=Crowell&rft.aufirst=Benjamin&rft_id=http%3A%2F%2Fwww.lightandmatter.com%2Fcalc%2F&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFGarrett2004" class="citation">Garrett, Paul (<time datetime="2004">2004</time>), <a rel="nofollow" class="external text" href="http://www.math.umn.edu/~garrett/calculus/"><i>Notes on First-Year Calculus</i></a>, <a href="/wiki/University_of_Minnesota" class="mw-redirect" title="University of Minnesota">University of Minnesota</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+First-Year+Calculus&rft.pub=University+of+Minnesota&rft.date=2004&rft.aulast=Garrett&rft.aufirst=Paul&rft_id=http%3A%2F%2Fwww.math.umn.edu%2F~garrett%2Fcalculus%2F&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFHussain2006" class="citation">Hussain, Faraz (<time datetime="2006">2006</time>), <a rel="nofollow" class="external text" href="http://www.understandingcalculus.com/"><i>Understanding Calculus</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Calculus&rft.date=2006&rft.aulast=Hussain&rft.aufirst=Faraz&rft_id=http%3A%2F%2Fwww.understandingcalculus.com%2F&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFKeisler2000" class="citation">Keisler, H. Jerome (<time datetime="2000">2000</time>), <a rel="nofollow" class="external text" href="http://www.math.wisc.edu/~keisler/calc.html"><i>Elementary Calculus: An Approach Using Infinitesimals</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Calculus%3A+An+Approach+Using+Infinitesimals&rft.date=2000&rft.aulast=Keisler&rft.aufirst=H.+Jerome&rft_id=http%3A%2F%2Fwww.math.wisc.edu%2F~keisler%2Fcalc.html&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFMauch2004" class="citation">Mauch, Sean (<time datetime="2004">2004</time>), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060415161115/http://www.its.caltech.edu/~sean/book/unabridged.html"><i>Unabridged Version of Sean's Applied Math Book</i></a>, arhivat din <a rel="nofollow" class="external text" href="http://www.its.caltech.edu/~sean/book/unabridged.html">original</a> la <time datetime="2006-04-15">15 aprilie 2006</time><span class="reference-accessdate">, accesat în <time datetime="2011-02-07">7 februarie 2011</time></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unabridged+Version+of+Sean%27s+Applied+Math+Book&rft.date=2004&rft.aulast=Mauch&rft.aufirst=Sean&rft_id=http%3A%2F%2Fwww.its.caltech.edu%2F~sean%2Fbook%2Funabridged.html&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFSloughter2000" class="citation">Sloughter, Dan (<time datetime="2000">2000</time>), <a rel="nofollow" class="external text" href="http://synechism.org/drupal/de2de/"><i>Difference Equations to Differential Equations</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Difference+Equations+to+Differential+Equations&rft.date=2000&rft.aulast=Sloughter&rft.aufirst=Dan&rft_id=http%3A%2F%2Fsynechism.org%2Fdrupal%2Fde2de%2F&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFStrang1991" class="citation">Strang, Gilbert (<time datetime="1991">1991</time>), <a rel="nofollow" class="external text" href="http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm"><i>Calculus</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.date=1991&rft.aulast=Strang&rft.aufirst=Gilbert&rft_id=http%3A%2F%2Focw.mit.edu%2Fans7870%2Fresources%2FStrang%2Fstrangtext.htm&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFStroyan1997" class="citation">Stroyan, Keith D. (<time datetime="1997">1997</time>), <a rel="nofollow" class="external text" href="http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm"><i>A Brief Introduction to Infinitesimal Calculus</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Brief+Introduction+to+Infinitesimal+Calculus&rft.date=1997&rft.aulast=Stroyan&rft.aufirst=Keith+D.&rft_id=http%3A%2F%2Fwww.math.uiowa.edu%2F~stroyan%2FInfsmlCalculus%2FInfsmlCalc.htm&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><cite id="CITEREFWikibooks" class="citation">Wikibooks, <a class="external text" href="https://en.wikibooks.org/wiki/Calculus"><i>Calculus</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.au=Wikibooks&rft_id=http%3A%2F%2Fen.wikibooks.org%2Fwiki%2FCalculus&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADerivat%C4%83" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> "<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Derivative.html">Derivative.</a>" From <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://easycalculation.com/differentiation/differentiation-calculator.php">Differentiation Calculator</a></li> <li><a rel="nofollow" class="external text" href="http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/trig2.html">Derivatives of Trigonometric functions</a>, UBC</li> <li><a rel="nofollow" class="external text" href="http://calculus.solved-problems.com/category/derivative/">Solved Problems in Derivatives</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130827111431/http://calculus.solved-problems.com/category/derivative/">Arhivat</a> în <time datetime="2013-08-27">27 august 2013</time>, la <a href="/wiki/Wayback_Machine" class="mw-redirect" title="Wayback Machine">Wayback Machine</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Adus de la <a dir="ltr" href="https://ro.wikipedia.org/w/index.php?title=Derivată&oldid=16228657">https://ro.wikipedia.org/w/index.php?title=Derivată&oldid=16228657</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categorii" title="Special:Categorii">Categorii</a>: <ul><li><a href="/wiki/Categorie:Analiz%C4%83_matematic%C4%83" title="Categorie:Analiză matematică">Analiză matematică</a></li><li><a href="/wiki/Categorie:Calcul_diferen%C8%9Bial" title="Categorie:Calcul diferențial">Calcul diferențial</a></li><li><a href="/w/index.php?title=Categorie:Func%C8%9Bii_%C8%99i_cartografieri&action=edit&redlink=1" class="new" title="Categorie:Funcții și cartografieri — pagină inexistentă">Funcții și cartografieri</a></li><li><a href="/w/index.php?title=Categorie:Operatori_liniari_%C3%AEn_calcul_diferen%C8%9Bial&action=edit&redlink=1" class="new" title="Categorie:Operatori liniari în calcul diferențial — pagină inexistentă">Operatori liniari în calcul diferențial</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie ascunsă: <ul><li><a href="/wiki/Categorie:Webarchive_template_wayback_links" title="Categorie:Webarchive template wayback links">Webarchive template wayback links</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Ultima editare a paginii a fost efectuată la 29 aprilie 2024, ora 14:07.</li> <li id="footer-info-copyright">Acest text este disponibil sub licența <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ro">Creative Commons cu atribuire și distribuire în condiții identice</a>; pot exista și clauze suplimentare. 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