{"id":41,"date":"2023-09-17T10:58:59","date_gmt":"2023-09-17T10:58:59","guid":{"rendered":"https:\/\/mathority.org\/de\/tangentengleichung\/"},"modified":"2023-09-17T10:58:59","modified_gmt":"2023-09-17T10:58:59","slug":"tangentengleichung","status":"publish","type":"post","link":"https:\/\/mathority.org\/de\/tangentengleichung\/","title":{"rendered":"Tangentengleichung"},"content":{"rendered":"<p>In diesem Artikel erfahren Sie <strong>, wie Sie die Gleichung der Tangente an eine Kurve finden<\/strong> . Dar\u00fcber hinaus k\u00f6nnen Sie mit gel\u00f6sten \u00dcbungen unterschiedlicher Schwierigkeitsgrade trainieren. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-de-la-recta-tangente-a-una-funcion-en-un-punto\"><\/span> Gleichung der Tangente an eine Funktion in einem Punkt <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px;\">\n<p style=\"text-align:left\"> Die <strong>Gleichung der Tangente<\/strong> an die Funktion f(x) im Punkt x=x <sub>0<\/sub> 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-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p align=\"LEFT\"> Wobei der Punkt P(x <sub>0<\/sub> ,y <sub>0<\/sub> ) der Punkt ist, an dem die Tangente und die Funktion zusammenfallen. Und die Steigung der Tangente m ist gleich der Ableitung der Kurve am Punkt x <sub>0<\/sub> , d. h. m=f'(x <sub>0<\/sub> ). <\/p>\n<\/div>\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\/equation-de-la-tangente-ligne.webp\" alt=\"Tangentengleichung\" class=\"wp-image-2306\" width=\"463\" height=\"461\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Im Bild oben sehen Sie eine Kurve<\/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-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> dargestellt in Blau und einer orangefarbenen Tangente an die Funktion<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Um<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd5304ac1643ba3660a7efa36ade1983_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"51\" style=\"vertical-align: -3px;\"><\/p>\n<p> , da sie nur diesen Punkt gemeinsam haben. Nun, die Gleichung dieser Tangente lautet<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<p> und seine Steigung ist<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-ecuacion-de-la-recta-tangente\"><\/span> So finden Sie die Tangentengleichung<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Um die Tangentengleichung an eine Funktion an einem Punkt zu finden, m\u00fcssen Sie Folgendes tun:<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Ermitteln Sie die Steigung der Tangente, indem Sie die Ableitung der Funktion am Tangentenpunkt berechnen.<\/span><\/li>\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Bestimmen Sie einen Punkt auf der Tangente.<\/span><\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\"><strong>Finden Sie die Gleichung der Tangente<\/strong> unter Verwendung der berechneten Steigung und des Punktes der Tangente.<\/span> <\/li>\n<\/ol>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-ecuacion-de-la-recta-tangente-a-una-curva\"><\/span> Beispiel f\u00fcr die Gleichung der Tangente an eine Kurve<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Nachdem wir die Theorie der Tangentengleichung kennengelernt haben, sehen wir uns an, wie man die Tangentengleichung berechnet, indem wir Schritt f\u00fcr Schritt ein Beispiel l\u00f6sen:<\/p>\n<ul>\n<li> Berechnen Sie die Gleichung der Tangente an die Kurve\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> Um<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/li>\n<\/ul>\n<p> Wir wissen, dass die Tangentengleichung immer die folgende Form hat:<\/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-326424811181144df35c0b94ce50c462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -y_0= m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Als erstes muss die Steigung der Geraden berechnet werden. Somit ist die Steigung der Tangente,<\/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-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , wird der Wert der Ableitung der Kurve am Tangentialpunkt x=1 sein, d. h<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a69005ee8bf2d80d73b989ad0cedccd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<p> Daher differenzieren wir die Funktion und berechnen dann <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb68a498d3bd60e51d3dc230691f886c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" 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-92c9a5ee4068789701733f793fbac622_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x \\quad \\longrightarrow \\quad f'(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"293\" 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-d96270e9b7d7c3cae8baea602cea53bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)= 2\\cdot 1+1=2+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" 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-2443a93bff265c6b8ba692ef8d14f633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(1)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sobald wir den Wert kennen<\/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-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , wir m\u00fcssen einen Punkt finden<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e531a80c13865d1ad612bd3f634efa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x_0,y_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -5px;\"><\/p>\n<p> der Tangente, um die Tangentengleichung zu vervollst\u00e4ndigen.<\/p>\n<p> Die <strong>Tangentengleichung und die Kurvengleichung haben immer einen gemeinsamen Punkt<\/strong> , der in diesem Fall der 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-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Daher wie die Kurve<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> durch diesen Punkt geht, k\u00f6nnen wir die andere Komponente des Punktes durch Berechnung ermitteln <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32045357853caad8774629c95963835d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" 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-7b2601f20b100f2635bc0342175b4627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^2+1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Der Tangentenpunkt ist 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-d26257abb9047188ab3e3887f447e20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sowohl die Kurve als auch die Tangente verlaufen durch diesen Punkt, sodass wir ihn auch verwenden k\u00f6nnen, um die Gleichung der Tangente zu finden.<\/p>\n<p> Es bleibt nur noch, die gefundenen Werte der Steigung und des Tangentenpunkts in seine Gleichung einzusetzen:<\/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-0321e19825c08a1f47a00b2cf625088f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=3 \\qquad P(1,2) \\end{array} \\right\\} \\longrightarrow \\ y -2= 3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kurz gesagt lautet die Tangentengleichung: <\/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-037e5c42a4adef3e5ba970a66b8d3459_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y-2=3(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background is-style-wide\" style=\"background-color:#1976d2;color:#1976d2\">\n<p> Sie k\u00f6nnen die Tangentengleichung auch mit der expliziten Geradengleichung ausdr\u00fccken: <\/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-59097f2ef899c7c608e2527467021b23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=3x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background is-style-wide\" style=\"background-color:#1976d2;color:#1976d2\">\n<p> Unten sehen Sie die dargestellte Kurve<\/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-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> und seine Tangente an <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2883f0b53c531552fde7ff189f83165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\"><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1020651b62576571e0ac9c0cb65dd287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2=3(x-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/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\/equation-de-la-tangente-a-une-courbe-en-un-point.webp\" alt=\"Gleichung einer Tangente an eine Kurve an einem Punkt\" class=\"wp-image-2318\" width=\"445\" height=\"434\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Wie Sie sehen k\u00f6nnen, die Kurve<\/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-ea8efcaacce7a90d3cc483105986c47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<p> und die Tangente<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3cb3f64ce416f3a5c6cc80c11cae9afb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-2=3(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<p> Sie haben nur den Punkt gemeinsam<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42a32282c97c3b8d9f90b2f1418844d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> , genau wie wir es berechnet haben. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-de-la-recta-tangente\"><\/span> Aufgaben zur Tangentengleichung gel\u00f6st<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> \u00dcbung 1<\/h3>\n<p> Berechnen Sie die Gleichung der Tangente an die Kurve<\/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-99f623c4fede3b664682c5cbc1aab81d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<p> Um <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-71e6033606cd14039ab202fb7a18c50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2 .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/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-left\"> Die Tangentengleichung hat immer die folgende Form:<\/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-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Schritt 1: Berechnen Sie die Steigung der Tangente<\/strong><\/p>\n<p class=\"has-text-align-left\"> Die Steigung <em>m<\/em> ist der Wert der Ableitung der Kurve am Tangentialpunkt. Daher in diesem Fall <\/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-c7192bf7bd4300d7d77fe084134d6849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = f'(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" 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-ab9e97205d5c9f2fbfcb085cbfdbdd75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3 \\ \\longrightarrow \\ f'(x)= 4x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"319\" 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-e3cb482f7e68631c8dcc5705ac1257d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2)= 4\\cdot 2-4=8-4=4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" 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-b0e7f5cd5a44a120769c9d3a1eae02c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(2)=4\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Schritt 2: Suchen Sie einen Punkt auf der Tangente<\/strong><\/p>\n<p class=\"has-text-align-left\"> Die Tangentengleichung und die Kurvengleichung haben immer einen gemeinsamen Punkt, der in diesem Fall der 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-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Daher wie die Kurve<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> durch diesen Punkt geht, k\u00f6nnen wir die andere Komponente des Punktes durch Berechnung ermitteln <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f026e401162db03299777455b748b308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-99f623c4fede3b664682c5cbc1aab81d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2-4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"156\" 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-67ff5b86b2842fefbdf2ddc7c2df39f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2\\cdot 2^2-4\\cdot 2+3 =2 \\cdot 4 -8 +3 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"326\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Somit ist der Punkt, durch den sowohl die Kurve als auch die Tangente verlaufen, der Punkt<\/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-d744fa34f41ed1bbe3fdf2c5ad7f55a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2,3).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Schritt 3: Schreiben Sie die Tangentengleichung<\/strong><\/p>\n<p class=\"has-text-align-left\"> Es bleibt nur noch, die gefundenen Werte der Steigung und des Tangentenpunkts in seine Gleichung einzusetzen:<\/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-1622c6ecd4d43bb4fc4901b437464652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=4 \\qquad P(2,3) \\end{array} \\right\\} \\longrightarrow \\ y -3= 4(x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Die Tangentengleichung lautet daher: <\/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-c1233301c390a75095fc24bd8765e081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y -3= 4(x-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 2<\/h3>\n<p> Berechnen Sie die Gleichung der Tangente an die Kurve<\/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-d1309dcf6b647174b562cb71ab600c1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=-3x^2+2x\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"139\" style=\"vertical-align: -5px;\"><\/p>\n<p> am Koordinatenursprung. <\/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-left\"> Der Koordinatenursprung bezieht sich auf den Punkt<\/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-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<p> Wir m\u00fcssen daher die Tangente an die Funktion im Punkt berechnen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-791f3561f68c75b943d5af446c9f988f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zuerst bestimmen wir den Wert der Steigung der Tangente, indem wir die Ableitung im Koordinatenursprung 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-e3f5e3bd3a06eb5e2831d90e5fc0f31d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-3x^2+2x \\ \\longrightarrow \\  f'(x)= -6x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"315\" 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-d07228d62a796c695cb75841830d0e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0)= -6\\cdot 0+2=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" 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-075eebc763002b84e54211e61242356f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(0)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In diesem Fall kennen wir bereits einen Punkt, durch den die Tangente verl\u00e4uft. Denn die Aussage sagt uns, dass die Gerade die Kurve im Koordinatenursprung, also im Punkt, tangieren muss<\/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-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Der Punkt, den die Kurve und die Tangente gemeinsam haben, ist also der Punkt<\/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-d5f3123d35179a39bd727675fca259c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zum Schluss setzen Sie einfach die gefundenen Werte f\u00fcr die Steigung und den Tangentenpunkt in Ihre Gleichung ein:<\/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-de8e4e9dbb7a5bca1d591612abcf7730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=2 \\qquad P(0,0) \\end{array} \\right\\} \\longrightarrow \\ y -0= 2(x-0)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zusammenfassend lautet die Tangentengleichung: <\/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-19b9a613ed0c41d6c98ab37c6a0a1331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -0= 2(x-0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" 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-329a7fc0d44a0b32cbb521e81ee50db6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = 2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 3<\/h3>\n<p> Berechnen Sie die Tangente an die Kurve<\/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-33130a168a4e20b536fb742b8ce2a662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-2x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -5px;\"><\/p>\n<p> was parallel zur rechten Seite ist<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4e35094bc85458a54e2b47228f9c39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -4px;\"><\/p>\n<p> . <\/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-left\"> In dieser Aufgabe wird uns gesagt, dass die Tangente parallel zur Geraden sein muss<\/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-02c5cab1d3747c5baa1ded66f3055f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<p> Und zwei Geraden sind parallel, wenn sie die gleiche Steigung haben. Die Tangente muss also die gleiche Steigung haben wie die Gerade<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3dee6df2aaaef2e062c41a79057de62e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Das bedeutet, dass wir die Steigung der Geraden ermitteln m\u00fcssen<\/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-02c5cab1d3747c5baa1ded66f3055f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 .\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dazu l\u00f6schen wir die Variable und:<\/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-6390ce01aebdda3a7305c4dd1e55d4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-4x-6=0 \\ \\longrightarrow \\ y =4x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"246\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Also die Steigung der Linie<\/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-824b16de72dde879834460d93bd88610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<p> ist 4, da die Steigung einer Geraden die Zahl ist, die x multipliziert, wenn y klar ist.<\/p>\n<p class=\"has-text-align-left\"> Daher muss die Steigung der Tangente auch 4 sein, denn damit sie parallel sind, m\u00fcssen sie die gleiche Steigung haben.<\/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-e5072b7479ca854c5e3cdea8ffff2c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In diesem Fall sagen sie uns nicht den Tangentenpunkt zwischen der Kurve und der Tangente. Aber wir wissen, dass die Ableitung der Kurve am Tangentenpunkt gleich der Steigung der Tangente ist, d. h<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Nun, wie k\u00f6nnen wir den Wert kennen?<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> , k\u00f6nnen wir x <sub>0<\/sub> aus der Gleichung ermitteln<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79089d702fc8e5b6b7342c0eb2f0c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dazu berechnen wir zun\u00e4chst die Ableitung von <\/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-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" 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-ac4abafb3c879a6fd9c906ff9eea94d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= x^2-2x-1 \\ \\longrightarrow \\ f'(x)=2x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"309\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und jetzt l\u00f6sen wir es<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> wissend, dass <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-421352ccc778c624805a5e2663bb7077_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/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-f99f5b23457229b93eb24c214942f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-6a9fc1e268bfa17b6fe04e5fafbaaedc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 =2(x_0)-2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" 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-74d7e6af89f911f5a194fee138e70afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4+2 =2x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"89\" style=\"vertical-align: -3px;\"><\/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-96c892854007fa06281252d3fcc3ae4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6 =2x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\"><\/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-9602ea885075b53499286a5126ad9724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{6}{2} =x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"49\" 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-f2fb028e2d1cb1011436226f865d5162_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und sobald wir die x-Koordinate des Punktes kennen, k\u00f6nnen wir durch Berechnung die andere Koordinate des Punktes ermitteln <\/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-4763abfc9310baf690c4bb81c5d8b743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" 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-9e36063894754424dc75ff41070c42ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=3^2-2\\cdot 3-1= 9-6-1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"281\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Somit ist der Punkt, durch den sowohl die Kurve als auch die Tangente verlaufen, der Punkt<\/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-a3558010337a7cdc27dddb44c10f0df1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Es bleibt nur noch, die gefundenen Werte der Steigung und des Tangentenpunkts in seine Gleichung einzusetzen:<\/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-8f1f49e9bef505c5c71cffd15f0d29d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=4 \\qquad P(3,2) \\end{array} \\right\\} \\longrightarrow \\ y -2= 4(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und die Tangentengleichung 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-89b6fd61abd22d30db13453334da7135_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y -2=4(x-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 4<\/h3>\n<p> Berechnen Sie die Tangente an die Kurve<\/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-ab7fb2898ec2a42b558f032b99518338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+5x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"155\" style=\"vertical-align: -5px;\"><\/p>\n<p> die mit der X-Achse einen Winkel von 45\u00b0 bildet. <\/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-left\"> Die Problemstellung sagt uns, dass die Tangente mit der X-Achse einen Winkel von 45\u00b0 bilden muss. In diesen F\u00e4llen muss die folgende Formel angewendet werden, um den Wert der Steigung zu ermitteln: <\/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-1e17ba52bc8d7a78aa6abe918856ba28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}\\left(\\alpha\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" 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-c891eec41f7529fbb36d622027b94d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\text{tg}\\left(45^{\\text{o}}\\right) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Die Anweisung gibt nicht den Tangentialpunkt zwischen der Kurve und der Tangente an. Aber wir wissen, dass die Ableitung der Kurve am Tangentenpunkt \u00e4quivalent zur Steigung der Tangente ist, d. h<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Wir k\u00f6nnen also x <sub>0<\/sub> berechnen, indem wir die Gleichung l\u00f6sen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79089d702fc8e5b6b7342c0eb2f0c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dazu berechnen wir zun\u00e4chst die Ableitung von <\/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-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" 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-d757979ec817338abf9a0d50e4d8838d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x^2+5x+1\\ \\longrightarrow \\ f'(x)=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"318\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und jetzt l\u00f6sen wir es<\/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-5c606eb4b268b71562672c32a0461053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m=f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<p> wissend, dass <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32704d9853b0093395b41eb385ebb4e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/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-f99f5b23457229b93eb24c214942f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m =f'(x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-eec283acc7af9f75a48ed262d785d7f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1 =4(x_0)+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" 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-4bcad96ea71d673fba2f814bffaee7c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1-5 =4x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\"><\/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-7638e49e3e5eab4f64f4dc439d458ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-4 =4x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"71\" style=\"vertical-align: -3px;\"><\/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-8561c7a3def54db216a8f1ebf2588e79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-4}{4} =x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"71\" 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-64d0311d2aa3328e9ed1f2073e90e4bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1=x_0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"62\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und sobald wir die x-Koordinate des Punktes kennen, k\u00f6nnen wir durch Berechnung die andere Koordinate des Punktes ermitteln <\/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-ab353a75f8c672950ea7d8376104722d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" 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-33f491d7f9d0eeeb767d846b5650734f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=2(-1)^2+5(-1)+1=2\\cdot 1  -5 + 1 = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"382\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">Somit ist der Punkt, durch den sowohl die Kurve als auch die Tangente verlaufen, der Punkt<\/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-d2ae436ead5cc58de912263561cfbe63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,-2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Es bleibt nur noch, die gefundenen Werte der Steigung und des Tangentenpunkts in seine Gleichung einzusetzen:<\/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-8ed772b3993de50c4c67631a6fd33040_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} y -y_0= m(x-x_0) \\\\[3ex] m=1 \\qquad P(-1,-2) \\end{array} \\right\\} \\longrightarrow \\ y -(-2)= 1(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"414\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich f\u00fchren wir die Operationen aus, um die Tangentengleichung zu finden: <\/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-28409d87564a3385166261f1fe92c01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y -(-2)=1(x-(-1))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" 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-9ebfc08d88b0dcf9c22ff7f225afdabf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y +2=1(x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" 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-b1c1b9eabdc99bc1ff5b5e8cdb5baf94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y + 2=x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In diesem Artikel erfahren Sie , wie Sie die Gleichung der Tangente an eine Kurve finden . Dar\u00fcber hinaus k\u00f6nnen Sie mit gel\u00f6sten \u00dcbungen unterschiedlicher Schwierigkeitsgrade trainieren. Gleichung der Tangente an eine Funktion in einem Punkt Die Gleichung der Tangente an die Funktion f(x) im Punkt x=x 0 lautet: Wobei der Punkt P(x 0 ,y &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/de\/tangentengleichung\/\"> <span class=\"screen-reader-text\">Tangentengleichung<\/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":[17],"tags":[],"class_list":["post-41","post","type-post","status-publish","format-standard","hentry","category-funktionsdarstellung"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Tangentengleichung -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/de\/tangentengleichung\/\" \/>\n<meta property=\"og:locale\" content=\"de_DE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Tangentengleichung -\" \/>\n<meta property=\"og:description\" content=\"In diesem Artikel erfahren Sie , wie Sie die Gleichung der Tangente an eine Kurve finden . Dar\u00fcber hinaus k\u00f6nnen Sie mit gel\u00f6sten \u00dcbungen unterschiedlicher Schwierigkeitsgrade trainieren. 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