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id="toc-Geskiedenis-sublist" class="vector-toc-list"> <li id="toc-Antieke_tydperk" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Antieke_tydperk"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Antieke tydperk</span> </div> </a> <ul id="toc-Antieke_tydperk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Middeleeue" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Middeleeue"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Middeleeue</span> </div> </a> <ul id="toc-Middeleeue-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moderne_tye" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moderne_tye"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Moderne tye</span> </div> </a> <ul id="toc-Moderne_tye-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fondasies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fondasies"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Fondasies</span> </div> </a> <ul id="toc-Fondasies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beginsels" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beginsels"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Beginsels</span> </div> </a> <button aria-controls="toc-Beginsels-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>Wissel Beginsels subafdeling</span> </button> <ul id="toc-Beginsels-sublist" class="vector-toc-list"> <li id="toc-Limiete_en_die_oneindig_klein" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limiete_en_die_oneindig_klein"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Limiete en die oneindig klein</span> </div> </a> <ul id="toc-Limiete_en_die_oneindig_klein-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differensiale_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differensiale_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Differensiale calculus</span> </div> </a> <ul id="toc-Differensiale_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Leibniznotasie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Leibniznotasie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Leibniznotasie</span> </div> </a> <ul id="toc-Leibniznotasie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integrale_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integrale_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Integrale calculus</span> </div> </a> <ul id="toc-Integrale_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fundamentele_stelling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamentele_stelling"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Fundamentele stelling</span> </div> </a> <ul id="toc-Fundamentele_stelling-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sien_ook" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sien_ook"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sien ook</span> </div> </a> <ul id="toc-Sien_ook-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="Inhoud" 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="Wissel die inhoudsopgawe" > <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">Wissel die inhoudsopgawe</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">Wiskundige analise</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="Gaan na 'n artikel in 'n ander taal. Beskikbaar in 114 tale" > <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-114" 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">114 tale</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Analysis" title="Analysis – Switserse Duits" lang="gsw" hreflang="gsw" data-title="Analysis" data-language-autonym="Alemannisch" data-language-local-name="Switserse Duits" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Anal%C3%ADs_matematica" title="Analís matematica – Aragonees" lang="an" hreflang="an" data-title="Analís matematica" data-language-autonym="Aragonés" data-language-local-name="Aragonees" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%95%E0%A4%B2%E0%A4%A8_%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="कलन शास्त्र – Angika" lang="anp" hreflang="anp" data-title="कलन शास्त्र" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="تحليل رياضي – Arabies" lang="ar" hreflang="ar" data-title="تحليل رياضي" data-language-autonym="العربية" data-language-local-name="Arabies" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="গাণিতিক বিশ্লেষণ – Assamees" lang="as" hreflang="as" data-title="গাণিতিক বিশ্লেষণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamees" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Anal%C3%ADs_matem%C3%A1ticu" title="Analís matemáticu – Asturies" lang="ast" hreflang="ast" data-title="Analís matemáticu" data-language-autonym="Asturianu" data-language-local-name="Asturies" 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/Riyazi_analiz" title="Riyazi analiz – Azerbeidjans" lang="az" hreflang="az" data-title="Riyazi analiz" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidjans" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D0%B1%D2%AF%D0%BB%D0%B5%D0%B3%D0%B5)" title="Анализ (математика бүлеге) – Baskir" lang="ba" hreflang="ba" data-title="Анализ (математика бүлеге)" data-language-autonym="Башҡортса" data-language-local-name="Baskir" 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%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Матэматычны аналіз – Belarussies" lang="be" hreflang="be" data-title="Матэматычны аналіз" data-language-autonym="Беларуская" data-language-local-name="Belarussies" 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%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Матэматычны аналіз – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматычны аналіз" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математически анализ – Bulgaars" lang="bg" hreflang="bg" data-title="Математически анализ" data-language-autonym="Български" data-language-local-name="Bulgaars" 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%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AC%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="गणितीय बिस्लेषण – Bhojpuri" lang="bh" hreflang="bh" data-title="गणितीय बिस्लेषण" 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%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="গাণিতিক বিশ্লেষণ – Bengaals" lang="bn" hreflang="bn" data-title="গাণিতিক বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="Bengaals" 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/Matemati%C4%8Dka_analiza" title="Matematička analiza – Bosnies" lang="bs" hreflang="bs" data-title="Matematička analiza" data-language-autonym="Bosanski" data-language-local-name="Bosnies" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica – Katalaans" lang="ca" hreflang="ca" data-title="Anàlisi matemàtica" data-language-autonym="Català" data-language-local-name="Katalaans" 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/%D8%B4%DB%8C%DA%A9%D8%A7%D8%B1%DB%8C%DB%8C_%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9%DB%8C" title="شیکاریی ماتماتیکی – Sorani" lang="ckb" hreflang="ckb" data-title="شیکاریی ماتماتیکی" data-language-autonym="کوردی" data-language-local-name="Sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Analisa" title="Analisa – Korsikaans" lang="co" hreflang="co" data-title="Analisa" data-language-autonym="Corsu" data-language-local-name="Korsikaans" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%A1_anal%C3%BDza" title="Matematická analýza – Tsjeggies" lang="cs" hreflang="cs" data-title="Matematická analýza" data-language-autonym="Čeština" data-language-local-name="Tsjeggies" 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%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D0%BF%D0%B0%D0%B9%C4%95)" title="Анализ (математика пайĕ) – Chuvash" lang="cv" hreflang="cv" data-title="Анализ (математика пайĕ)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" 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/Dadansoddiad_mathemategol" title="Dadansoddiad mathemategol – Wallies" lang="cy" hreflang="cy" data-title="Dadansoddiad mathemategol" data-language-autonym="Cymraeg" data-language-local-name="Wallies" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Matematisk_analyse" title="Matematisk analyse – Deens" lang="da" hreflang="da" data-title="Matematisk analyse" data-language-autonym="Dansk" data-language-local-name="Deens" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Analysis" title="Analysis – Duits" lang="de" hreflang="de" data-title="Analysis" data-language-autonym="Deutsch" data-language-local-name="Duits" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Analizo_matematik%C3%AAn" title="Analizo matematikên – Zazaki" lang="diq" hreflang="diq" data-title="Analizo matematikên" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7" title="Μαθηματική ανάλυση – Grieks" lang="el" hreflang="el" data-title="Μαθηματική ανάλυση" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis – Engels" lang="en" hreflang="en" data-title="Mathematical analysis" data-language-autonym="English" data-language-local-name="Engels" 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/Analitiko" title="Analitiko – Esperanto" lang="eo" hreflang="eo" data-title="Analitiko" 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/An%C3%A1lisis_matem%C3%A1tico" title="Análisis matemático – Spaans" lang="es" hreflang="es" data-title="Análisis matemático" data-language-autonym="Español" data-language-local-name="Spaans" 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/Matemaatiline_anal%C3%BC%C3%BCs" title="Matemaatiline analüüs – Estnies" lang="et" hreflang="et" data-title="Matemaatiline analüüs" data-language-autonym="Eesti" data-language-local-name="Estnies" 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/Analisi_matematiko" title="Analisi matematiko – Baskies" lang="eu" hreflang="eu" data-title="Analisi matematiko" data-language-autonym="Euskara" data-language-local-name="Baskies" 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/%D8%A2%D9%86%D8%A7%D9%84%DB%8C%D8%B2_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="آنالیز ریاضی – Persies" lang="fa" hreflang="fa" data-title="آنالیز ریاضی" data-language-autonym="فارسی" data-language-local-name="Persies" 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/Analyysi_(matematiikka)" title="Analyysi (matematiikka) – Fins" lang="fi" hreflang="fi" data-title="Analyysi (matematiikka)" data-language-autonym="Suomi" data-language-local-name="Fins" 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/Analyse_(math%C3%A9matiques)" title="Analyse (mathématiques) – Frans" lang="fr" hreflang="fr" data-title="Analyse (mathématiques)" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Analysis" title="Analysis – Noord-Fries" lang="frr" hreflang="frr" data-title="Analysis" data-language-autonym="Nordfriisk" data-language-local-name="Noord-Fries" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anail%C3%ADs_mhatamaitici%C3%BAil" title="Anailís mhatamaiticiúil – Iers" lang="ga" hreflang="ga" data-title="Anailís mhatamaiticiúil" data-language-autonym="Gaeilge" data-language-local-name="Iers" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E5%88%86%E6%9E%90" title="數學分析 – Gan-Sjinees" lang="gan" hreflang="gan" data-title="數學分析" data-language-autonym="贛語" data-language-local-name="Gan-Sjinees" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Analiz_(mat%C3%A9matik)" title="Analiz (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Analiz (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Anailis_mhatamataigeach" title="Anailis mhatamataigeach – Skotse Gallies" lang="gd" hreflang="gd" data-title="Anailis mhatamataigeach" data-language-autonym="Gàidhlig" data-language-local-name="Skotse Gallies" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_matem%C3%A1tica" title="Análise matemática – Galisies" lang="gl" hreflang="gl" data-title="Análise matemática" data-language-autonym="Galego" data-language-local-name="Galisies" 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%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית – Hebreeus" lang="he" hreflang="he" data-title="אנליזה מתמטית" data-language-autonym="עברית" data-language-local-name="Hebreeus" 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%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="गणितीय विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="गणितीय विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis – Fiji Hindi" lang="hif" hreflang="hif" data-title="Mathematical analysis" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dka_analiza" title="Matematička analiza – Kroaties" lang="hr" hreflang="hr" data-title="Matematička analiza" data-language-autonym="Hrvatski" data-language-local-name="Kroaties" 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/Matematikai_anal%C3%ADzis" title="Matematikai analízis – Hongaars" lang="hu" hreflang="hu" data-title="Matematikai analízis" data-language-autonym="Magyar" data-language-local-name="Hongaars" 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/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6" title="Մաթեմատիկական անալիզ – Armeens" lang="hy" hreflang="hy" data-title="Մաթեմատիկական անալիզ" data-language-autonym="Հայերեն" data-language-local-name="Armeens" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Analyse_(mathematica)" title="Analyse (mathematica) – Interlingua" lang="ia" hreflang="ia" data-title="Analyse (mathematica)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_matematis" title="Analisis matematis – Indonesies" lang="id" hreflang="id" data-title="Analisis matematis" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesies" 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/Analitiko" title="Analitiko – Ido" lang="io" hreflang="io" data-title="Analitiko" 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/St%C3%A6r%C3%B0fr%C3%A6%C3%B0igreining" title="Stærðfræðigreining – Yslands" lang="is" hreflang="is" data-title="Stærðfræðigreining" data-language-autonym="Íslenska" data-language-local-name="Yslands" 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/Analisi_matematica" title="Analisi matematica – Italiaans" lang="it" hreflang="it" data-title="Analisi matematica" data-language-autonym="Italiano" data-language-local-name="Italiaans" 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/%E8%A7%A3%E6%9E%90%E5%AD%A6" title="解析学 – Japannees" lang="ja" hreflang="ja" data-title="解析学" data-language-autonym="日本語" data-language-local-name="Japannees" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Matimatikal_analisis" title="Matimatikal analisis – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Matimatikal analisis" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%9C%E1%83%90%E1%83%9A%E1%83%98%E1%83%96%E1%83%98" title="მათემატიკური ანალიზი – Georgies" lang="ka" hreflang="ka" data-title="მათემატიკური ანალიზი" data-language-autonym="ქართული" data-language-local-name="Georgies" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D1%82%D0%B0%D0%BB%D0%B4%D0%B0%D1%83" title="Математикалық талдау – Kazaks" lang="kk" hreflang="kk" data-title="Математикалық талдау" data-language-autonym="Қазақша" data-language-local-name="Kazaks" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%B6%E0%B3%8D%E0%B2%B2%E0%B3%87%E0%B2%B7%E0%B2%A3_%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ವಿಶ್ಲೇಷಣ ಗಣಿತ – Kannada" lang="kn" hreflang="kn" data-title="ವಿಶ್ಲೇಷಣ ಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EC%84%9D%ED%95%99_(%EC%88%98%ED%95%99)" title="해석학 (수학) – Koreaans" lang="ko" hreflang="ko" data-title="해석학 (수학)" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%82%D0%B0%D0%BB%D0%B4%D0%BE%D0%BE" title="Математикалык талдоо – Kirgisies" lang="ky" hreflang="ky" data-title="Математикалык талдоо" data-language-autonym="Кыргызча" data-language-local-name="Kirgisies" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Analysis_mathematica" title="Analysis mathematica – Latyn" lang="la" hreflang="la" data-title="Analysis mathematica" data-language-autonym="Latina" data-language-local-name="Latyn" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Analys_(Mathematik)" title="Analys (Mathematik) – Luxemburgs" lang="lb" hreflang="lb" data-title="Analys (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxemburgs" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Analise_matematical" title="Analise matematical – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Analise matematical" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Analixi_matematica" title="Analixi matematica – Ligurian" lang="lij" hreflang="lij" data-title="Analixi matematica" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Analisi_matematega" title="Analisi matematega – Lombard" lang="lmo" hreflang="lmo" data-title="Analisi matematega" 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%81%E0%BA%B2%E0%BA%99%E0%BA%A7%E0%BA%B4%E0%BB%80%E0%BA%84%E0%BA%B2%E0%BA%B0%E0%BA%97%E0%BA%B2%E0%BA%87%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94" title="ການວິເຄາະທາງຄະນິດສາດ – Lao" lang="lo" hreflang="lo" data-title="ການວິເຄາະທາງຄະນິດສາດ" data-language-autonym="ລາວ" data-language-local-name="Lao" 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/Matematin%C4%97_analiz%C4%97" title="Matematinė analizė – Litaus" lang="lt" hreflang="lt" data-title="Matematinė analizė" data-language-autonym="Lietuvių" data-language-local-name="Litaus" 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/Matem%C4%81tisk%C4%81_anal%C4%ABze" title="Matemātiskā analīze – Letties" lang="lv" hreflang="lv" data-title="Matemātiskā analīze" data-language-autonym="Latviešu" data-language-local-name="Letties" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Математичка анализа – Masedonies" lang="mk" hreflang="mk" data-title="Математичка анализа" data-language-autonym="Македонски" data-language-local-name="Masedonies" 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%A8%E0%B4%BE%E0%B4%B2%E0%B4%BF%E0%B4%B8%E0%B4%BF%E0%B4%B8%E0%B5%8D_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="അനാലിസിസ് (ഗണിതം) – Malabaars" lang="ml" hreflang="ml" data-title="അനാലിസിസ് (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="Malabaars" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Analisis_matematik" title="Analisis matematik – Maleis" lang="ms" hreflang="ms" data-title="Analisis matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Maleis" 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/Analisi_matematika" title="Analisi matematika – Maltees" lang="mt" hreflang="mt" data-title="Analisi matematika" data-language-autonym="Malti" data-language-local-name="Maltees" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/An%C3%A1leze_matem%C3%A1tica" title="Análeze matemática – Mirandees" lang="mwl" hreflang="mwl" data-title="Análeze matemática" data-language-autonym="Mirandés" data-language-local-name="Mirandees" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%81%E1%80%BD%E1%80%B2%E1%80%81%E1%80%BC%E1%80%99%E1%80%BA%E1%80%B8%E1%80%85%E1%80%AD%E1%80%90%E1%80%BA%E1%80%96%E1%80%BC%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="ခွဲခြမ်းစိတ်ဖြာသင်္ချာ – Birmaans" lang="my" hreflang="my" data-title="ခွဲခြမ်းစိတ်ဖြာသင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Birmaans" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Analysis" title="Analysis – Nederduits" lang="nds" hreflang="nds" data-title="Analysis" data-language-autonym="Plattdüütsch" data-language-local-name="Nederduits" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Analyse_(wiskunde)" title="Analyse (wiskunde) – Nederlands" lang="nl" hreflang="nl" data-title="Analyse (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Nederlands" 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/Matematisk_analyse" title="Matematisk analyse – Nuwe Noors" lang="nn" hreflang="nn" data-title="Matematisk analyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Nuwe Noors" 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/Matematisk_analyse" title="Matematisk analyse – Boeknoors" lang="nb" hreflang="nb" data-title="Matematisk analyse" data-language-autonym="Norsk bokmål" data-language-local-name="Boeknoors" 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/Analisi_matematica" title="Analisi matematica – Oksitaans" lang="oc" hreflang="oc" data-title="Analisi matematica" data-language-autonym="Occitan" data-language-local-name="Oksitaans" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4_%E0%A8%B5%E0%A8%BF%E0%A8%B8%E0%A8%BC%E0%A8%B2%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%A3" title="ਗਣਿਤ ਵਿਸ਼ਲੇਸ਼ਣ – Pandjabi" lang="pa" hreflang="pa" data-title="ਗਣਿਤ ਵਿਸ਼ਲੇਸ਼ਣ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Pandjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_matematyczna" title="Analiza matematyczna – Pools" lang="pl" hreflang="pl" data-title="Analiza matematyczna" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/An%C3%A0lisi_matem%C3%A0tica" title="Anàlisi matemàtica – Piedmontese" lang="pms" hreflang="pms" data-title="Anàlisi matemàtica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DB%8C%D8%AA%DA%BE%D9%85%DB%8C%D9%B9%DB%8C%DA%A9%D9%84_%D8%A7%D9%86%DB%8C%D9%84%DB%8C%D8%B3%D8%B2" title="میتھمیٹیکل انیلیسز – Western Punjabi" lang="pnb" hreflang="pnb" data-title="میتھمیٹیکل انیلیسز" 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/An%C3%A1lise_matem%C3%A1tica" title="Análise matemática – Portugees" lang="pt" hreflang="pt" data-title="Análise matemática" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Analiza_matematic%C4%83" title="Analiza matematică – Roemeens" lang="ro" hreflang="ro" data-title="Analiza matematică" data-language-autonym="Română" data-language-local-name="Roemeens" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_(%D1%80%D0%B0%D0%B7%D0%B4%D0%B5%D0%BB_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B8)" title="Анализ (раздел математики) – Russies" lang="ru" hreflang="ru" data-title="Анализ (раздел математики)" data-language-autonym="Русский" data-language-local-name="Russies" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D1%96%D1%87%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7%D0%B0" title="Математічна аналіза – Rusyn" lang="rue" hreflang="rue" data-title="Математічна аналіза" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/An%C3%A0lisi_(matim%C3%A0tica)" title="Anàlisi (matimàtica) – Sisiliaans" lang="scn" hreflang="scn" data-title="Anàlisi (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sisiliaans" 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/Mathematical_analysis" title="Mathematical analysis – Skots" lang="sco" hreflang="sco" data-title="Mathematical analysis" data-language-autonym="Scots" data-language-local-name="Skots" 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/Matemati%C4%8Dka_analiza" title="Matematička analiza – Serwo-Kroaties" lang="sh" hreflang="sh" data-title="Matematička analiza" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serwo-Kroaties" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%B8%E0%B6%BA_%E0%B7%80%E0%B7%92%E0%B7%81%E0%B7%8A%E0%B6%BD%E0%B7%9A%E0%B7%82%E0%B6%AB%E0%B6%BA" title="ගණිතමය විශ්ලේෂණය – Sinhala" lang="si" hreflang="si" data-title="ගණිතමය විශ්ලේෂණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical analysis" 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/Matematick%C3%A1_anal%C3%BDza" title="Matematická analýza – Slowaaks" lang="sk" hreflang="sk" data-title="Matematická analýza" data-language-autonym="Slovenčina" data-language-local-name="Slowaaks" 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/Matemati%C4%8Dna_analiza" title="Matematična analiza – Sloweens" lang="sl" hreflang="sl" data-title="Matematična analiza" data-language-autonym="Slovenščina" data-language-local-name="Sloweens" 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/Analiza_matematikore" title="Analiza matematikore – Albanees" lang="sq" hreflang="sq" data-title="Analiza matematikore" data-language-autonym="Shqip" data-language-local-name="Albanees" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Математичка анализа – Serwies" lang="sr" hreflang="sr" data-title="Математичка анализа" data-language-autonym="Српски / srpski" data-language-local-name="Serwies" 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/Matematisk_analys" title="Matematisk analys – Sweeds" lang="sv" hreflang="sv" data-title="Matematisk analys" data-language-autonym="Svenska" data-language-local-name="Sweeds" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Uchambuzi_wa_kihisabati" title="Uchambuzi wa kihisabati – Swahili" lang="sw" hreflang="sw" data-title="Uchambuzi wa kihisabati" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="பகுவியல் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="பகுவியல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A7%E0%B8%B4%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B2%E0%B8%B0%E0%B8%AB%E0%B9%8C" title="คณิตวิเคราะห์ – Thai" lang="th" hreflang="th" data-title="คณิตวิเคราะห์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Analiz" title="Analiz – Turkmeens" lang="tk" hreflang="tk" data-title="Analiz" data-language-autonym="Türkmençe" data-language-local-name="Turkmeens" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pagsusuring_matematikal" title="Pagsusuring matematikal – Tagalog" lang="tl" hreflang="tl" data-title="Pagsusuring matematikal" 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/Analiz_(matematik)" title="Analiz (matematik) – Turks" lang="tr" hreflang="tr" data-title="Analiz (matematik)" data-language-autonym="Türkçe" data-language-local-name="Turks" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Vukambisisi_bya_Dyondzo-Tinhlayo" title="Vukambisisi bya Dyondzo-Tinhlayo – Tsonga" lang="ts" hreflang="ts" data-title="Vukambisisi bya Dyondzo-Tinhlayo" data-language-autonym="Xitsonga" data-language-local-name="Tsonga" class="interlanguage-link-target"><span>Xitsonga</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математик анализ – Tataars" lang="tt" hreflang="tt" data-title="Математик анализ" data-language-autonym="Татарча / tatarça" data-language-local-name="Tataars" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Математичний аналіз – Oekraïens" lang="uk" hreflang="uk" data-title="Математичний аналіз" data-language-autonym="Українська" data-language-local-name="Oekraïens" 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/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AA%D8%AD%D9%84%DB%8C%D9%84" title="ریاضیاتی تحلیل – Oerdoe" lang="ur" hreflang="ur" data-title="ریاضیاتی تحلیل" data-language-autonym="اردو" data-language-local-name="Oerdoe" 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/Matematik_analiz" title="Matematik analiz – Oezbeeks" lang="uz" hreflang="uz" data-title="Matematik analiz" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Oezbeeks" 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/An%C3%A0%C5%82ixi_matem%C3%A0tica" title="Anàłixi matemàtica – Venetian" lang="vec" hreflang="vec" data-title="Anàłixi matemàtica" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Analiz_(matematikan_jaguz)" title="Analiz (matematikan jaguz) – Veps" lang="vep" hreflang="vep" data-title="Analiz (matematikan jaguz)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%E1%BA%A3i_t%C3%ADch_to%C3%A1n_h%E1%BB%8Dc" title="Giải tích toán học – Viëtnamees" lang="vi" hreflang="vi" data-title="Giải tích toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Viëtnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Analisis_matematikal" title="Analisis matematikal – Waray" lang="war" hreflang="war" data-title="Analisis matematikal" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90" title="数学分析 – Wu-Sjinees" lang="wuu" hreflang="wuu" data-title="数学分析" data-language-autonym="吴语" data-language-local-name="Wu-Sjinees" 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%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A9%D7%A2%D7%A8_%D7%90%D7%A0%D7%90%D7%9C%D7%99%D7%96" title="מאטעמאטישער אנאליז – Jiddisj" lang="yi" hreflang="yi" data-title="מאטעמאטישער אנאליז" data-language-autonym="ייִדיש" data-language-local-name="Jiddisj" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%8Ct%C3%BAw%C3%B2_Mathim%C3%A1t%C3%AD%C3%ACk%C3%AC" title="Ìtúwò Mathimátíìkì – Yoruba" lang="yo" hreflang="yo" data-title="Ìtúwò Mathimátíìkì" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90" title="数学分析 – Chinees" lang="zh" hreflang="zh" data-title="数学分析" data-language-autonym="中文" data-language-local-name="Chinees" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%88%86%E6%9E%90%E5%AD%B8" title="分析學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="分析學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E5%88%86%E6%9E%90" title="數學分析 – Kantonees" lang="yue" hreflang="yue" data-title="數學分析" data-language-autonym="粵語" data-language-local-name="Kantonees" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q7754#sitelinks-wikipedia" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">versteek</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">in Wikipedia, die vrye ensiklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="af" dir="ltr"><p><b>Wiskundige analise</b>, ook algemeen bekend as <b>calculus</b> (van die Latynse word <i>calculus</i> wat letterlik “klein klippie wat gebruik word om op te tel” beteken) is die wiskundige studie van aanhoudende verandering, in dieselfde manier wat geometrie die studie van vorm is en algebra die studie van die veralgemenings van rekeningkundige berekeninge is. Dit het twee hoof vertakkings, naamlik: differensiale calculus (wat die tempo van verandering en die helling van kurwes meet) en integrale calculus (wat die akkumulasie van hoeveelhede en die areas onder en tussen kurwes meet).  Hierdie twee vertakkings is verwant aan mekaar deur die fundamentele teorie van calculus. Beide vertakkings maak gebruik van die fundamentele notasies van konvergensie van die oneindigende reekse tot ŉ goed- gedefinieerde limiet. Oor die algemeen word aanvaar dat moderne calculus in die 17de eeu ontwikkel is deur <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> en <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Leibniz</a>. Vandag word calculus in ’n wye verskeidenheid velde gebruik; soos die <a href="/wiki/Wetenskap" title="Wetenskap">wetenskap</a>, <a href="/wiki/Ingenieurswese" title="Ingenieurswese">ingenieurswese</a> en <a href="/wiki/Ekonomie" title="Ekonomie">ekonomie</a>. </p><p>Calculus is deel van moderne wiskundige opvoeding. ŉ Kursus in calculus word beskou as die begin tot ander, meer gevorderde kursusse in wiskunde.. Calculus het histories bekend gestaan as “die calculus van infinitesimale”, of “infinitesimale calculus”. <i>Calculus</i> is ook gebruik vir die benaming van sommige metodes van berekeninge soos variasie calculus, lambda calculus en proses calculus<b>.</b> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Geskiedenis">Geskiedenis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=1" title="Wysig afdeling: Geskiedenis" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=1" title="Edit section's source code: Geskiedenis"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Moderne calculus is in die <a href="/wiki/17de_eeu" title="17de eeu">17de eeu</a> in <a href="/wiki/Europa" title="Europa">Europa</a> ontwikkel deur Isaac Newton en Gottfried Wilhelm Leibniz (onafhanklik van mekaar, maar het kort na mekaar hul werke oor calculus gepubliseer).  Maar elemente daarvan het reeds in <a href="/wiki/Antieke_Griekeland" title="Antieke Griekeland">antieke Griekeland</a> en later ook in <a href="/wiki/China" class="mw-disambig" title="China">China</a>, die <a href="/wiki/Midde-Ooste" title="Midde-Ooste">Middel Ooste</a>, weer in Europa in die Middeleeue en in <a href="/wiki/Indi%C3%AB" title="Indië">Indië</a> verskyn. </p> <div class="mw-heading mw-heading3"><h3 id="Antieke_tydperk">Antieke tydperk</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=2" title="Wysig afdeling: Antieke tydperk" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=2" title="Edit section's source code: Antieke tydperk"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die antieke periode het sommige idees bekendgestel wat tot integrale calculus gelei het, maar dit wil voorkom of hierdie idees in ŉ rigiede en sistematiese wyse ontwikkel is nie. Berekeninge van volume en area, een doelwit van integrale calculus, kan aangetref word in die Egiptiese Moskou papirusse (1820 v.c.) maar die formules is eenvoudig met belangrike komponente wat nie daar is nie en daar is ook geen definitiewe metode ten opsigte van die gebruik van die formules nie.  Sedert die eeu van Griekse wiskundiges (408-355 v.c.) het Eudoxus die metode van uitputting gebruik (wat die voorloper van die konsep van die limiet was) om areas en oppervlakte te bereken.  <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (287-212 v.c.) het hierdie idee verder ontwikkel en het heuristiek, wat dieselfde metodes van integrale calculus bevat ontwikkel. Die metode van uitputting is later ook op ŉ onafhanklike wyse in China deur Liu Hui in die 3de eeu na Christus ontdek en is gebruik om die area van ŉ sirkel te bepaal.  In die 5de eeu na Christus het Zu Gengzhi, die seun van Zu Chongzhi ŉ metode ontwikkel wat later bekend sou staan as die Cavalieri prinsiep- om die volume van ŉ sfeer te bepaal. </p> <div class="mw-heading mw-heading3"><h3 id="Middeleeue">Middeleeue</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=3" title="Wysig afdeling: Middeleeue" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=3" title="Edit section's source code: Middeleeue"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In die Midde-Ooste het <a href="/wiki/Alhazen" title="Alhazen">Alhazen</a> (965-1040 n.c.) ŉ formule ontwikkel om die som van die vierde kragte te bepaal.  Hy het die resultate gebruik om om wat nou bekend staan as integrasie van die funksie, waar die formule vir die som van die integrale vierkante en vierde magte hom toegelaat het om die volume van ŉ parabool te bereken.  In die 14de eeu het Indiese wiskundiges nie-rigiede metodes van ŉ soort van differensiasie van sommige trigotomies funksies gegee.  Madhava van Sangamagrama en die Kerala skool van astronomie en wiskunde het daarvolgens die komponente van calculus bepaal.  ŉ Volledige teorie wat hierdie komponente bevat is nou wel bekend in die Westerse wêreld as die Taylor-reeks of die nimmereindigende reeks aannames.  Tog kon hierdie wiskundiges nie die verskillende idees kombineer om die twee onderliggende temas van afleiding en integrasie te wys, te konnekteer en in die wonderlike probleemoplossing metodes wat ons vandag het verander het nie. </p> <div class="mw-heading mw-heading3"><h3 id="Moderne_tye">Moderne tye</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=4" title="Wysig afdeling: Moderne tye" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=4" title="Edit section's source code: Moderne tye"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <blockquote><p>  “Die calculus was die eerste bereiking van die moderne wiskunde en dit is moeilik om die belangrikheid daarvan te oorskat.  Ek dink dit definieer meer onomwonde as enige iets anders die aanvangs van die moderne wiskunde en sistematiese wiskundige analiste wat die logiese ontwikkeling daarvan is.  Dit bewys die grootste tegniese vooruitgang in presiese denke.” –John von Neumann.</p></blockquote><p>In Europa is die fundamentele werk geboekstaaf as gevolg van <a href="/wiki/Bonaventura_Cavalieri" title="Bonaventura Cavalieri">Bonaventura Cavalieri</a> wat geargumenteer het dat volumes en areas as die som van die volumes en areas van infinitesimale dun kruis-lyne bereken moet word.  Hierdie idees was eenders aan die van Archimedes in sy <i>The Method</i>.  Hierdie werk het egter verlore geraak in die <a href="/wiki/13de_eeu" title="13de eeu">13de eeu</a> en is eers weer in die <a href="/wiki/20ste_eeu_v.C." title="20ste eeu v.C.">20ste</a> eeu herontdek en was gevolglik onbekend aan Cavalieri.  Cavalieri se werk word nie op ag geslaan nie, omdat sy metodes tot verkeerde resultate kan lei en die infinitesimale hoeveelhede wat hy voorgestel het, debatteerbaar was. </p><p>Die formele studie van calculus het Cavalieri se infinitesimales met die calculus van finiete veranderinge wat in ongeveer die selfde tyd in Europa ontwikkel is, bymekaar gebring. <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> het voorgehou dat hy idees van <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> geleen het toe hy die konsep van <i>adequality</i> (wat die konsep van gelykheid tot ŉ infinitesimale fout-terme) ontwikkel het.  John Wallis, Isaac Newton en <a href="/wiki/James_Gregory" title="James Gregory">James Gregory</a> het daarin geslaag om die kombinasie wat die tweede fundamentele teorie van calculus gevorm het, bymekaar te voeg. Die produksiereël en kettingreël, die notasie van hoër afleiers, Taylor-reeks en analitiese funksies is deur Isaac Newton in ŉ idiosinkratiese notasie bekend gestel wat hy gebruik het om probleme van wiskundige fisika op te los.  In sy werke het Newton sy idees omskryf om by die wiskundige idiome van die tyd aan te pas.  Hy het sy metodes van calculus gebruik om oplossings tot verskeie probleme te kry, byvoorbeeld: Beweging van planete, die vorm van die oppervlak ŉ roterende vloeistof, die vorm van die Aarde en die beweging van ŉ gewig wat binne ŉ silinder beweeg.  Dit word volledig in sy werk <i>Principia Mathematica</i> (<a href="/wiki/1687" title="1687">1687</a>) omskryf.  </p><p>Isaac Newton se idees is verder ontwikkel tot ŉ volledige calculus van infinitesimales deur Gottfried Wilhelm Leibniz (wat eers deur Newton van plagiaat beskuldig is). Leibniz word nou beskou as ŉ onafhanklike ontwikkelaar en medewerker tot calculus. Sy grootste bydrae tot calculus was om ŉ definitiewe stel reëls te ontwikkel.  Anders as Newton het Leibniz ook baie aandag aan formalisasie geskenk en het baie tyd daaraan spandeer om die regte simbole vir die regte konsepte te ontwikkel. </p><p>Leibniz en Newton word gevolglik beide gekrediteer met die ontwikkeling van calculus.  Newton was die eerste wat calculus op fisika toegepas het terwyl Leibniz baie meer van die notasie wat ons in moderne calculus gebruik ontwikkel het.  </p><p>Tydens die publikasie van Newton en Leibniz se werke was daar baie kontroversie oor watter wiskundige (en gevolglik watter land) die eer moet verdien.  Hoewel Newton eerste sy resultate ontwikkel het, het Leibniz eerste gepubliseer en Newton het gevolglik gesê dat Leibniz idees uit sy ongepubliseerde notas gesteel het.  Dit het tot jarelange onmin tussen <a href="/wiki/Engelse" title="Engelse">Engelse</a>- en Kontinentale Europese wiskundiges gelei. ŉ Ondersoek in die werke van hierdie twee wiskundiges wys egter dat al twee op ŉ onafhanklike wyse hul resultate verkry het.  Leibniz het eerste met integrasie begin terwyl Newton eerste met differensiasie begin het.  Vandag word beide erken as die ontwikkelaars van calculus, maar dit is Leibniz wat die nuwe dissipline sy naam gegee het aangesien Newton dit die “Studie van fluktuasies” genoem het.  </p><p>Na Leibniz en Newton het verskeie wiskundiges bydraes gelewer tot die studie van calculus.  Een van die eerste en mees volledige werke oor beide infinitesimale en integrale wiskunde is geskryf deur Maria Gaetana Agnesi in <a href="/wiki/1748" title="1748">1748</a> </p> <div class="mw-heading mw-heading2"><h2 id="Fondasies">Fondasies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=5" title="Wysig afdeling: Fondasies" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=5" title="Edit section's source code: Fondasies"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In calculus verwys fondasie na die rigiede ontwikkeling van die onderwerp van aksiome en definisies. In vroeë calculus is daar geglo dat die gebruik van infinitesimale hoeveelhede nie rigied was nie, en is hewig deur verskeie outeurs gekritiseer.  Michelle Rolle en Bishop Berkeley is twee van die outeurs.  Berkeley het infinitesimales beskryf as die “spoke van verdwynende kwantiteite” is sy <i>The Analyst</i> (1734).  Om ŉ rigiede fondasie vir calculus te ontwikkel het baie wiskundiges vir die grootste gedeelte van die eeu na Leibniz en Newton besig gehou.  Dit is tot vandag nog ŉ aktiewe deel van calculus-navorsing. </p><p>Verskeie wiskundiges, insluitend Maclaurin het probeer om die stabiliteit in die gebruik van infinitesimales te ontwikkel, maar dit sal nie tot 150 jaar later met die verskyning van die werk van Cauchy en Weierstrass wees wat daar ŉ manier gevind is om die notasies van klein hoeveelhede te vermy nie.   </p> <div class="mw-heading mw-heading2"><h2 id="Beginsels">Beginsels</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=6" title="Wysig afdeling: Beginsels" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=6" title="Edit section's source code: Beginsels"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Limiete_en_die_oneindig_klein">Limiete en die oneindig klein</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=7" title="Wysig afdeling: Limiete en die oneindig klein" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=7" title="Edit section's source code: Limiete en die oneindig klein"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analise word gewoonlik ontwikkel deur baie klein hoeveelhede te manipuleer. Die eerste metode om hierdie mee te bewerkstellig was, histories, deur die gebruik van <i>infinitesimale</i>. Hierdie "voorwerpe", wat aangewend word asof hulle getalle is wat bloot "oneindig klein" is. 'n Infinitesimale getal <i>dx</i> sou groter as 0 wees, maar kleiner as enige getal in die ry </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,{\frac {1}{2}},{\frac {1}{3}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,{\frac {1}{2}},{\frac {1}{3}},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e6ac038114a374e417333a3bec164c34124810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.985ex; height:5.176ex;" alt="{\displaystyle 1,{\frac {1}{2}},{\frac {1}{3}},\ldots }"></span></dd></dl> <p>en minder as enige positiewe reële getal. Enige heelgetalveelvoud van 'n infinitesimaal is steeds oneindig klein, m.a.w. infinitesimale gehoorsaam nie die <a href="/w/index.php?title=Archimediese_eienskap&action=edit&redlink=1" class="new" title="Archimediese eienskap (bladsy bestaan nie)">Archimediese eienskap</a> nie. </p><p>Vanuit hierdie oogpunt, is calculus 'n versameling tegnieke wat gemoeid is op die manipulering van oneindige klein getalle. Hierdie benadering het egter teen die einde van die 19de eeu in onbruik verval, omdat dit moeilik was om die idee van 'n infinitesimaal nougeset te definieer. Ten spyte hiervan is die konsep weer in die 20ste eeu in die lewe geroep met die inlywing van <a href="/w/index.php?title=Nie-standaard_analise&action=edit&redlink=1" class="new" title="Nie-standaard analise (bladsy bestaan nie)">nie-standaard calculus</a> en <a href="/w/index.php?title=Egalige_infinitesimale_analise&action=edit&redlink=1" class="new" title="Egalige infinitesimale analise (bladsy bestaan nie)">egalige infinitesimale calculus</a>, wat die weg gebaan het vir wiskundig nougesette manipulering van infinitesimale. </p><p>Infinitesimale is tydens die 19de eeu vervang deur <a href="/wiki/Limiete" class="mw-redirect" title="Limiete">limiete</a>. Limiete beskryf die waarde van 'n wiskundige <a href="/wiki/Funksie" title="Funksie">funksie</a> by 'n sekere punt "invoer" in terme van waardes by "invoere" wat in die omliggende area is. Hulle beskryf funksiegedrag op 'n baie klein skaal, maar gebruik die gewone reële getalsisteem. </p><p>In hierdie opsig is analise 'n versameling van tegnieke om sekere limiete te manipuleer. Infinitesimale word vervang deur baie klein getalle, en die oneindige klein gedrag van 'n funksie word gevind deur die beperkende gedrag vir toenemend kleiner getalle te neem. Limiete is maklik om op 'n nougesette basis te sit en om hierdie rede word hulle gewoonlik beskou as die standaardbenadering tot analise. </p> <div class="mw-heading mw-heading3"><h3 id="Differensiale_calculus">Differensiale calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=8" title="Wysig afdeling: Differensiale calculus" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=8" title="Edit section's source code: Differensiale calculus"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/L%C3%AAer:Tangent_derivative_calculusdia-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Tangent_derivative_calculusdia-2.svg/300px-Tangent_derivative_calculusdia-2.svg.png" decoding="async" width="300" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Tangent_derivative_calculusdia-2.svg/450px-Tangent_derivative_calculusdia-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Tangent_derivative_calculusdia-2.svg/600px-Tangent_derivative_calculusdia-2.svg.png 2x" data-file-width="561" data-file-height="387" /></a><figcaption>Raaklyn by (<i>x</i>; <i>f</i>(<i>x</i>)). Die afgeleide <i>f′</i>(<i>x</i>) van 'n kurwe by 'n punt is die gradiënt (verandering in funksiewaarde oor invoerwaarde) van die raaklyn aan die kurwe by daai punt.</figcaption></figure> <p>Differensiale calculus is die studie van die definisie, eienskappe en toepassings van die <a href="/wiki/Afgeleide" title="Afgeleide">afgeleide</a> van 'n <a href="/wiki/Funksie" title="Funksie">funksie</a>. Die proses waardeur die afgeleide gevind word, is <i>differensiasie</i>. Gegee 'n funksie en 'n punt in sy definisieversameling, dan beskryf die afgeleide by daardie punt die klein-skaal gedrag van die funksie naby daardie punt. Deur die afgeleide van 'n funksie by elke punt in sy definiesieversameling te vind, word dit moontlik om 'n nuwe funksie, genaamd die <i>afgeleide funksie</i>, te kry. In wiskundige <a href="/wiki/Jargon" title="Jargon">jargon</a> is die afgeleide 'n <a href="/w/index.php?title=Line%C3%AAre_operator&action=edit&redlink=1" class="new" title="Lineêre operator (bladsy bestaan nie)">lineêre operator</a> wat 'n funksie ingevoer word en 'n tweede funksie uitvoer. </p><p>Die algemeenste simbool vir 'n afgeleide is die apostroofagtige merk genaamd <a href="/w/index.php?title=Priem_(simbool)&action=edit&redlink=1" class="new" title="Priem (simbool) (bladsy bestaan nie)">priem</a>. Die afgeleide van die funksie <i>f</i> is dus <i>f′</i>. Byvoorbeeld, as <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> die kwadraatfunksie waarvan die afgeleide die verdubbelingsfunksie is, kan die afgeleide met die volgende notasie aangedui word: <i>f′</i>(<i>x</i>) = 2<i>x</i>. </p><p>As die invoerwaarde tyd is, verteenwoordig die afgeleide verandering met betrekking tot tyd. Byvoorbeeld, as <i>f</i> a funksie is wat tyd neem as invoer en die posisie van 'n bal by 'n sekere tydstip as uitvoer gee, verteenwoordig die afgeleide van <i>f</i> hoe die posisie met betrekking tot tyd verander, d.w.s. die <a href="/wiki/Snelheid" title="Snelheid">snelheid</a> van die bal. </p><p>As die funksie lineêr is (m.a.w. die grafiek van die funksie is 'n reguitlyn), kan die funksie geskryf word as <i>y</i> = <i>mx</i> + <i>b</i>, waar: </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 m={{\mbox{verandering in }}y \over {\mbox{verandering in }}x}={\Delta y \over {\Delta x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>verandering in </mtext> </mstyle> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>verandering in </mtext> </mstyle> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={{\mbox{verandering in }}y \over {\mbox{verandering in }}x}={\Delta y \over {\Delta x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd77d11e4803ca43b9de70bf31a2d0dc86066b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.217ex; height:6.009ex;" alt="{\displaystyle m={{\mbox{verandering in }}y \over {\mbox{verandering in }}x}={\Delta y \over {\Delta x}}.}"></span></dd></dl> <p>Dit gee die presies waarde van die gradiënt van 'n reguitlyn. As die grafiek van die funksie egter nie 'n reguitlyn is nie, sal die verandering in <i>y</i> gedeel deur die verandering in <i>x</i> afwissel. Afgeleides verleen 'n presiese betekenis tot die idee van verandering in uitvoer met betrekking tot verandering in invoer. Om konkreet te wees, laat <i>f</i> 'n funksie wees, en stel 'n punt <i>a</i> in die definisieversameling van <i>f</i> vas. (<i>a</i>; <i>f</i>(<i>a</i>)) is 'n punt op die grafiek van die funksie. As <i>h</i> 'n getal baie naby aan nul is, dan is <i>a</i> + <i>h</i> 'n getal baie naby aan <i>a</i>. Dus is (<i>a</i> + <i>h</i>; <i>f</i>(<i>a</i> + <i>h</i>)) baie naby aan (<i>a</i>; <i>f</i>(<i>a</i>)). Die gradiënt tussen hierdie twee punte is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2017cd12adcbfa0f07bcc1090006133781fff8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.867ex; height:6.509ex;" alt="{\displaystyle m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}.}"></span></dd></dl> <p>Hierdie uitdrukking word genoem 'n <i>differensiekwosiënt</i>. 'n Lyn deur twee punte op 'n kurwe word genoem 'n <i><a href="/wiki/Snylyn" title="Snylyn">snylyn</a></i>, so <i>m</i> is die gradiënt van die snylyn tussen (<i>a</i>; <i>f</i>(<i>a</i>)) en (<i>a</i> + <i>h</i>, <i>f</i>(<i>a</i> + <i>h</i>)). Die snylyn is net 'n benadering tot die gedrag van die funksie by die punt <i>a</i>, omdat dit nie rekening hou van wat tussen <i>a</i> en <i>a</i> + <i>h</i> met die funksie gebeur nie. Verder is dit ook onmoontlik om uit te vind wat die gedrag by <i>a</i> is deur <i>h</i> gelyk te stel aan nul, omdat dit deling deur nul noodsaak, wat ontoelaatbaar is. Die afgeleide is dus gedefinieer deur die <a href="/wiki/Limiet" title="Limiet">limiet</a> te neem soos <i>h</i> streef na nul, wat beteken dat dit die gedrag van <i>f</i> vir alle klein waardes van <i>h</i> oorweeg en 'n bestendige waarde ontgin vir die geval waar <i>h</i> gelyk is aan nul: </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 \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3d79e9ff8acf531bb7596a6e0687fbb8027ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.937ex; height:5.843ex;" alt="{\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.}"></span></dd></dl> <p>Meetkundig gesproke is die afgeleide die gradiënt van die <a href="/w/index.php?title=Raaklyn&action=edit&redlink=1" class="new" title="Raaklyn (bladsy bestaan nie)">raaklyn</a> tot die grafiek van <i>f</i> by <i>a</i>. Die raaklyn is 'n limiet van snylyne net soos die afgeleide 'n limiet van differensiekwosiënte is. Om hierdie rede word die afgeleide soms die gradiënt van die funksie <i>f</i> genoem. </p><p>Hier is 'n spesifieke voorbeeld -- die afgeleide van die kwadraatsfunksie by invoer 3. Laat <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> die kwadraatsfunksie wees. </p><p><br /> </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 {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>9</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>+</mo> <mn>6</mn> <mi>h</mi> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>9</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>h</mi> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>6</mn> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>6.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e137fbcaa67304926deb682a9684ebf79aabf1d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:29.203ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-9 \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6.\end{aligned}}}"></span></dd></dl> <p>Die gradiënt van die raaklyn tot 'n kwadreringsfunksie by die punt (3;9) is 6. D.w.s. die funksiewaarde vermeerder ses keer vinniger as die invoerwaarde. Die limietproses wat sopas beskryf is kan toegepas word vir enige punt in die definisieversameling van die kwadreringsfunksie. Dit definieer die <i>afgeleide funksie</i> van die kwadreringsfunksie. </p> <div class="mw-heading mw-heading3"><h3 id="Leibniznotasie">Leibniznotasie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=9" title="Wysig afdeling: Leibniznotasie" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=9" title="Edit section's source code: Leibniznotasie"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>'n Algemene notasie vir die afgeleide in die voorbeeld hier bo is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}y=x^{2}\\{\frac {dy}{dx}}=2x.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y=x^{2}\\{\frac {dy}{dx}}=2x.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d187fff30b41554baf4938d00b04e2ef33770d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:10.371ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}y=x^{2}\\{\frac {dy}{dx}}=2x.\end{aligned}}}"></span></dd></dl> <p>Hierdie notasie is die eerste keer deur Leibniz gebruik en dra dus sy naam. </p><p>In 'n benardering gebaseer op limiete moet die simbool <i>dy/dx</i> nie geïnterpreteer word as die kwosiënt van die twee getalle nie, maar as beknopte voorstelling van die limiet wat hier bo uitgewerk is. Leibniz se oorspronklike bedoeling was egter dat dit die kwosiënt van twee oneindig klein getalle voorstel, waar <i>dy</i> die infinitesimale verandering in <i>y</i> is wat veroorsaak word deur 'n infinitesimale verandering <i>dx</i> in <i>x</i>. </p><p>Ons kan ook aan <i>d/dx</i> dink as die differensiasie-operator wat 'n funksie neem as invoer en nog 'n funksie, die afgeleide, as uitvoer lewer. Byvoorbeeld: </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 {d}{dx}}(x^{2})=2x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}(x^{2})=2x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4fd3f805c38024f3887477b61fd2297f4e8050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.812ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}(x^{2})=2x.}"></span></dd></dl> <p>Soos dit hier gebruik word, beteken die <i>dx</i> in die noemer "met betrekking tot <i>x</i>". Selfs as analise ontwikkel word deur limiete te gebruik in plaas van infinitesimale, is dit steeds algemeen om simbole soos <i>dx</i> en <i>dy</i> te manipuleer asof hulle regte getalle is en, hoewel dit moontlik is om sulke manipulering te vermy, is dit soms gerieflik om die notasies van sekere operasies so uit te druk. </p> <div class="mw-heading mw-heading3"><h3 id="Integrale_calculus">Integrale calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=10" title="Wysig afdeling: Integrale calculus" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=10" title="Edit section's source code: Integrale calculus"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fundamentele_stelling">Fundamentele stelling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=11" title="Wysig afdeling: Fundamentele stelling" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=11" title="Edit section's source code: Fundamentele stelling"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die <i>fundamentele stelling van die calculus</i> stel dat differensiasie en integrasie inverse operasies is. Om meer presies te wees, beskryf dit die verband tussen die waardes van anti-afgeleides en bepaalde integrale. Omdat dit gewoonlik makliker is om 'n anti-afgeleide te bereken as wat dit is om die definisie van 'n bepaalde integraal toe te pas, verskaf die fundamentele stelling van kalkulus 'n praktiese metode om 'n bepaalde integraal te bereken. </p><p>Die fundamentele stelling van calculus stel: As 'n funksie <i>f</i> kontinu is oor die interval [<i>a</i>,<i>b</i>] en as <i>F</i> 'n funksie is waarvan die afgeleide <i>f</i> is oor die interval (<i>a</i>, <i>b</i>), dan </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 \int _{a}^{b}f(x)\,dx=F(b)-F(a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f470e7743fda04c3d353a4dee2f441ae454f528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.052ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}"></span></dd></dl> <p>Verder, vir elke <i>x</i> in die interval (<i>a</i>, <i>b</i>), </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 {d}{dx}}\int _{a}^{x}f(t)\,dt=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> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4f64cc882e88a4d7b634c37b7c1684630c3687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.325ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}"></span></dd></dl> <p>Hierdie besef, wat deur beide <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> en <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a> gemaak is, was onmisbaar in die ontsaglike vermenigvuldiging van analitiese resultate wat gelewer is nadat hulle werk bekend geword het. Die fundamentele stelling verskaf 'n algebraïese metode om verskeie bepaalde integrale te bereken — sonder om limietprosesse (soos Riemann-somme) toe te pas — deur formules vir anti-afgeleides te vind. </p> <div class="mw-heading mw-heading2"><h2 id="Sien_ook">Sien ook</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiskundige_analise&veaction=edit&section=12" title="Wysig afdeling: Sien ook" class="mw-editsection-visualeditor"><span>wysig</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Wiskundige_analise&action=edit&section=12" title="Edit section's source code: Sien ook"><span>wysig bron</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Numeriese_analise" title="Numeriese analise">Numeriese analise</a></li></ul> <div class="notice metadata plainlinks" id="saadjie"><span typeof="mw:File"><a href="/wiki/Wikipedia:Saadjie" title="Wikipedia:Saadjie"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/45px-Wiki_letter_w.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/68px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/90px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span> <i>Hierdie artikel is ’n <a href="/wiki/Wikipedia:Saadjie" title="Wikipedia:Saadjie">saadjie</a>. 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