{"id":34,"date":"2023-09-17T11:02:19","date_gmt":"2023-09-17T11:02:19","guid":{"rendered":"https:\/\/mathority.org\/de\/ableitung-des-arkustangens-1\/"},"modified":"2023-09-17T11:02:19","modified_gmt":"2023-09-17T11:02:19","slug":"ableitung-des-arkustangens-1","status":"publish","type":"post","link":"https:\/\/mathority.org\/de\/ableitung-des-arkustangens-1\/","title":{"rendered":"Ableitung des arkustangens"},"content":{"rendered":"<p>In diesem Artikel erfahren Sie, wie Sie den Arkustangens einer Funktion ableiten. Dar\u00fcber hinaus k\u00f6nnen Sie Beispiele f\u00fcr diese Art der Ableitung sehen und sogar mit gel\u00f6sten \u00dcbungen zur Ableitung des Arkustangens \u00fcben. Abschlie\u00dfend zeigen wir Ihnen noch den Beweis der Formel f\u00fcr die Ableitung des Arkustangens. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-derivada-del-arcotangente\"><\/span> Was ist die Ableitung des Arkustangens?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Die Ableitung des Arkustangens von x ist eins \u00fcber eins plus x zum Quadrat.<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdeb5e29b862b8b9d5bc9f4c2c747106_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{1+x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"404\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Daher ist die <strong>Ableitung des Arkustangens einer Funktion<\/strong> gleich dem Quotienten aus der Ableitung dieser Funktion geteilt durch eins plus dem Quadrat der Funktion.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6546edbf0ff2d0ccba20a7fac11b89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"404\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> In diesem Fall wurde die Funktion durch au dargestellt, also w\u00e4re dies die Formel f\u00fcr die Ableitung des Arkustangens der Funktion u. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-arctangente.webp\" alt=\"abgeleitet vom Arkustangens\" class=\"wp-image-1997\" width=\"389\" height=\"296\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Wie Sie sehen k\u00f6nnen, ist die Formel f\u00fcr die Ableitung des Umkehrtangens den Formeln f\u00fcr die Ableitungen von Arkussinus und Arkuskosinus sehr \u00e4hnlich. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-del-arcotangente\"><\/span> Beispiele f\u00fcr die Ableitung des Arkustangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Sobald wir die Formel f\u00fcr die Ableitung des Arkustangens kennen, erkl\u00e4ren wir die Ableitung mehrerer Beispiele dieser Art trigonometrischer Ableitungen. Auf diese Weise k\u00f6nnen Sie leichter verstehen, wie der Arkustangens einer Funktion abgeleitet wird. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-del-arcotangente-de-2x\"><\/span> Beispiel 1: Ableitung des Arkustangens von 2x<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c8877ac889f77baa22f66d4b2568418_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Wir wenden die Formel an, um die Ableitung zu l\u00f6sen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6546edbf0ff2d0ccba20a7fac11b89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"404\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Die Ableitung von 2x ist 2, also ist die Arkustangens-Ableitung von 2x 2 \u00fcber eins plus 2x im Quadrat: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2a5ff151a4471bb769c46ac896ee0ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(2x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2}{1+(2x)^2}}=\\cfrac{2}{1+ 4x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"518\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-del-arcotangente-de-x-al-cuadrado\"><\/span> Beispiel 2: Ableitung des Arkustangens von x im Quadrat<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3f62897c299972bd734ffe87b6d28e84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Um das Ergebnis der Ableitung dieses Beispiels zu finden, m\u00fcssen wir die Formel f\u00fcr die Ableitung des Arkustangens verwenden, die lautet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6546edbf0ff2d0ccba20a7fac11b89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"404\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Somit ist die Ableitung der Funktion x <sup>2<\/sup> 2x, also ist die Ableitung des Arkustangens von x hoch 2: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4518d6b8df16464b2a763eb7d736504d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(x^2) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{2x}{1+\\left(x^2\\right)^2}=\\cfrac{2x}{1+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"507\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-del-arcotangente-del-seno-de-x\"><\/span> Beispiel 3: Ableitung des Arkustangens des Sinus von x<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72df0a729eefc917694a84ecccd4a959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}\\bigl(\\text{sen}(x)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"170\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Um die Ableitung zu berechnen, m\u00fcssen Sie logischerweise die entsprechende Formel anwenden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c6546edbf0ff2d0ccba20a7fac11b89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}(u) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{u'}{1+u^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"404\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> In diesem Fall haben wir eine zusammengesetzte Funktion, also m\u00fcssen wir die Kettenregel anwenden, um die Ableitung des Arkustangens zu berechnen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3897b362bb6b4681404918f45e91565_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{arctan}\\bigl(\\text{sen}(x)\\bigr) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{\\text{cos}(x)}{1+\\text{sen}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"482\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-del-arcotangente\"><\/span> Gel\u00f6ste \u00dcbungen zur Ableitung des Arkustangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Leiten Sie die folgenden Arkustangensfunktionen her: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23cb449bba097b71c6154e6bfd755940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=\\text{arctan}(x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"164\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ec295f2cfb72911775d2bc47d379e11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=\\cfrac{\\text{arctan}(3x^4)}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"175\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e54c5dc5ee8464186009da410740df5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f(x)=\\text{arctan}(x^5-3x^3+10)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"252\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b63b4e04e749c7b7d4cfecca4391eee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) }f(x)=\\text{arctan}^3(4x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"181\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c72a415d8c2e54870c4dc2e92344cef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) }f(x)=\\text{arctan}\\bigl(\\ln(x)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"185\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4fad4c5f97c1492b8ad5df06b165d2c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f(x)=\\text{arctan}\\left(\\sqrt{x^2+2x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"227\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Sehen Sie sich die L\u00f6sung an<\/strong> <\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-effa4065c7c98ae655b2cc5bdf14ca07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=\\cfrac{3x^2}{1+\\left(x^3\\right)^2}=\\cfrac{3x^2}{1+x^6}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"235\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2de05c8d708d379982abc8461f5d8706_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=\\cfrac{12x^3}{2\\left(1+\\left(3x^4\\right)^2\\right)}=\\cfrac{6x^3}{1+9x^8}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"285\" style=\"vertical-align: -30px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d72faae19f9b5cd7a8d53364bcf9817a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f'(x)=\\cfrac{5x^4-9x^2}{1+\\left(x^5-3x^3+10\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"248\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea7d5ff105b7432c2756cdcbf44e311b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=3\\text{arctan}^2(4x^2)\\cdot \\cfrac{8x}{1+\\left(4x^2\\right)^2}=\\cfrac{24x\\cdot\\text{arctan}^2(4x^2)}{1+16x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"453\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6eefb865fce5124f8326d122437c3124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=\\cfrac{\\cfrac{1}{x}}{1+\\bigl(\\ln(x)\\bigr)^2}=\\cfrac{1}{x\\left(1+\\ln^2(x)\\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"315\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6faff8aba659922b2cfc784a4f3dae4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f'(x)=\\cfrac{1}{1+\\left(\\sqrt{x^2+2x}\\right)^2}\\cdot \\cfrac{2x+2}{2\\sqrt{x^2+2x}}=\\cfrac{x+1}{\\left(1+x^2+2x\\right)\\sqrt{x^2+2x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"524\" style=\"vertical-align: -33px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-formula-de-la-derivada-del-arcotangente\"><\/span>Demonstration der Formel f\u00fcr die Ableitung des Arkustangens<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Als n\u00e4chstes beweisen wir die Formel f\u00fcr die Ableitung des Arkustangens.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88c05a50eddb183a57270676d6ebc5cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\text{arctan}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Wir wandeln zun\u00e4chst den Arkustangens in einen Tangens um und machen uns dabei die Tatsache zunutze, dass der Arkustangens die Umkehrfunktion des Tangens ist:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e806041dd9dbc7cf01bb34014aa18d59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\text{tan}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Wir unterscheiden die beiden Seiten der Gleichung:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca9f080338b9ab014e272f81395146ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1=\\cfrac{1}{\\text{cos}^2(y)}\\cdot y'\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"114\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Wir l\u00f6schen und&#8216;:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05c04d0ce365d8bd7848d4923038778c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\text{cos}^2(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"91\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Andererseits wissen wir dank der grundlegenden trigonometrischen Identit\u00e4t, dass die Summe der Quadrate von Sinus und Cosinus gleich 1 ist. Wir k\u00f6nnen daher den vorherigen Ausdruck in einen Bruch umwandeln:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-34f7d2ec2a5836c843db8adea73d021f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}^2(y)+\\text{cos}^2(y)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5314e15fe7e8ff7c9155906e7725483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{\\text{cos}^2(y)}{1}=\\cfrac{\\text{cos}^2(y)}{\\text{sen}^2(y)+\\text{cos}^2(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"252\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Wir dividieren alle Terme durch das Quadrat des Kosinus:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-39e03a87b9a62ab2db6a56192e44f531_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{\\cfrac{\\text{cos}^2(y)}{\\text{cos}^2(y)}}{\\cfrac{\\text{sen}^2(y)}{\\text{cos}^2(y)}+\\cfrac{\\text{cos}^2(y)}{\\text{cos}^2(y)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"96\" width=\"176\" style=\"vertical-align: -44px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3cf88afedfff6a5c53ffb31df510b4ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{\\cfrac{\\text{sen}^2(y)}{\\text{cos}^2(y)}+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"70\" width=\"128\" style=\"vertical-align: -44px;\"><\/p>\n<\/p>\n<p> Der Sinus dividiert durch den Cosinus ist gleich dem Tangens, also:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dbf6d65fa67f0a2161bd99ee7431f015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db914ed4a068a6dff2598b981b1682d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{\\text{tan}^2(y)+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"126\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Wie wir oben gesehen haben, ist der Tangens \u00e4quivalent zur Variablen x, daher k\u00f6nnen wir den Ausdruck ersetzen, um die Formel f\u00fcr die Ableitung des Arkustangens zu erhalten:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61292316edd6cef99a6135989713cd22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y'=\\cfrac{1}{x^2+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"88\" style=\"vertical-align: -14px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In diesem Artikel erfahren Sie, wie Sie den Arkustangens einer Funktion ableiten. Dar\u00fcber hinaus k\u00f6nnen Sie Beispiele f\u00fcr diese Art der Ableitung sehen und sogar mit gel\u00f6sten \u00dcbungen zur Ableitung des Arkustangens \u00fcben. Abschlie\u00dfend zeigen wir Ihnen noch den Beweis der Formel f\u00fcr die Ableitung des Arkustangens. Was ist die Ableitung des Arkustangens? Die Ableitung &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/de\/ableitung-des-arkustangens-1\/\"> <span class=\"screen-reader-text\">Ableitung des arkustangens<\/span> Weiterlesen &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[6],"tags":[],"class_list":["post-34","post","type-post","status-publish","format-standard","hentry","category-derivate"],"yoast_head":"<!-- This site is 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