{"id":1722,"date":"2019-11-19T11:45:00","date_gmt":"2019-11-19T10:45:00","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1722"},"modified":"2019-11-19T11:45:02","modified_gmt":"2019-11-19T10:45:02","slug":"introduction-au-nombre-derive","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2019\/11\/19\/introduction-au-nombre-derive\/","title":{"rendered":"Introduction au nombre d\u00e9riv\u00e9"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Regardez cette animation :<\/p>\n\n\n\n<figure class=\"wp-block-video\"><video height=\"606\" style=\"aspect-ratio: 910 \/ 606;\" width=\"910\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2019\/11\/nombre-derive.mp4\"><\/video><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">N&#8217;est-elle pas belle ?<\/p>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">Outre le fait qu&#8217;elle soit pas trop moche, elle est lourde de sens&#8230; Explications demand\u00e9es !<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/11\/19\/introduction-au-nombre-derive\/#Definition_du_nombre_derive\" >D\u00e9finition du nombre d\u00e9riv\u00e9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/11\/19\/introduction-au-nombre-derive\/#Un_exemple_pour_calculer_un_nombre_derive\" >Un exemple pour calculer un nombre d\u00e9riv\u00e9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2019\/11\/19\/introduction-au-nombre-derive\/#Calcul_dune_limite_en_Terminale\" >Calcul d&#8217;une limite en Terminale<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Definition_du_nombre_derive\"><\/span>D\u00e9finition du nombre d\u00e9riv\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re une fonction <em>f<\/em> dont la courbe repr\u00e9sentative est not\u00e9e \\(C_f\\). On prend alors un point A (d&#8217;abscisse <em>a<\/em>) sur cette courbe, donc de coordonn\u00e9es (<em>a<\/em> ; <em>f<\/em>(<em>a<\/em>)).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, prenons un point M sur cette m\u00eame courbe qui a une abscisse <em>x<\/em> variable. On fait donc bouger M sur la courbe. Si on observe la droite (AM) quand M se rapproche de A, on constate qu&#8217;elle se rapproche d&#8217;une position bien particuli\u00e8re : on arrive presque \u00e0 une droite qui fr\u00f4le la courbe. On appelle cette droite la <em>tangente \u00e0 la courbe au point A<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, si on regarde le coefficient directeur de (AM), on a: $$\\frac{y_M-y_A}{x_M-x_A}$$ d&#8217;apr\u00e8s la formule vue en Seconde, ce qui donne:$$\\frac{f(x)-f(a)}{x-a}.$$Ainsi, d&#8217;apr\u00e8s l&#8217;observation graphique faite pr\u00e9c\u00e9demment, on peut dire que ce nombre se rapproche du coefficient directeur de la tangente \u00e0 la courbe au point A. Et bien, c&#8217;est ce nombre que l&#8217;on va d\u00e9finir comme le <em>nombre d\u00e9riv\u00e9 de f en a<\/em>, et on va le noter <em>f&#8217;<\/em>(<em>a<\/em>). On note alors:$$\\lim\\limits_{x\\to a}\\frac{f(x)-f(a)}{x-a}=f'(a).$$ <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut aussi voir les choses l\u00e9g\u00e8rement diff\u00e9remment : si on note <em>h<\/em> = <em>x<\/em> &#8211; <em>a<\/em>, c&#8217;est-\u00e0-dire l&#8217;\u00e9cart entre <em>x<\/em> et <em>a<\/em> (sans valeur absolue, on dit que l&#8217;\u00e9cart est <em>relatif<\/em>) alors si <em>x<\/em> se rapproche de <em>a<\/em>, cela signifie que <em>h<\/em> se rapproche de 0. De plus, on peut \u00e9crire <em>x<\/em> = <em>a<\/em> + <em>h<\/em> et la d\u00e9finition du nombre d\u00e9riv\u00e9 devient alors:$$\\lim\\limits_{h\\to0}\\frac{f(a+h)-f(a)}{h}=0.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En 1\u00e8re sp\u00e9cialit\u00e9 Math, on utilise plut\u00f4t la deuxi\u00e8me \u00e9criture. Cependant, en Terminale, la premi\u00e8re \u00e9criture est assez utile pour calculer certaines limites de fonctions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_exemple_pour_calculer_un_nombre_derive\"><\/span>Un exemple pour calculer un nombre d\u00e9riv\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Prenons la fonction \\(f(x)=\\sqrt{x}\\) et d\u00e9terminons son nombre d\u00e9riv\u00e9 en <em>a<\/em> = 2. Pour cela, on commence par exprimer en fonction de <em>h<\/em> et <em>a<\/em> le taux d&#8217;accroissement en <em>a<\/em>, c&#8217;est-\u00e0-dire le coefficient directeur de la droite (AM) que l&#8217;on a vue dans l&#8217;animation pr\u00e9c\u00e9dente.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\frac{f(a+h)-f(a)}{h}=\\frac{\\sqrt{2+h}-\\sqrt2}{h}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&#8217;id\u00e9e est de trouver vers quel nombre se rapproche ce quotient. Malheureusement, en l&#8217;\u00e9tat, on ne peut pas car le num\u00e9rateur se rapproche de 0 et il en est de m\u00eame pour le d\u00e9nominateur&#8230; Et on ne sait pas vers quel nombre se rapproche une fraction qui elle-m\u00eame se rapproche de &#8220;\\(\\frac{0}{0}\\)&#8221; (les guillemets sont importants car bien s\u00fbr, cette fraction n&#8217;existe pas). On dit que c&#8217;est une <em>ind\u00e9termination<\/em>. Il faut donc la <em>lever<\/em>. Pour cela, nous allons multiplier le num\u00e9rateur et le d\u00e9nominateur du quotient par l&#8217;<em>expression conjugu\u00e9e<\/em> du num\u00e9rateur (c&#8217;est-\u00e0-dire que l&#8217;on multiplie par la m\u00eame chose en rempla\u00e7ant le signe du milieu par son oppos\u00e9). On obtient alors:$$ \\begin{align*}\\frac{f(a+h)-f(a)}{h}&amp;=\\frac{\\sqrt{2+h}-\\sqrt2}{h}\\times\\frac{\\sqrt{2+h}+\\sqrt2}{ \\sqrt{2+h}+\\sqrt2}\\\\&amp;=\\frac{2+h-2}{h(  \\sqrt{2+h}+\\sqrt2)}\\\\&amp;=\\frac{h}{h(\\sqrt{2+h}+\\sqrt2)}\\\\&amp;=\\frac{1}{ \\sqrt{2+h}+\\sqrt2 }\\text{ pour }h\\neq0.\\end{align*}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ce qu&#8217;il y  d&#8217;int\u00e9ressant dans le fait de multiplier par l&#8217;expression conjugu\u00e9e, c&#8217;est de faire appara\u00eetre au num\u00e9rateur une identit\u00e9 remarquable de la forme :$$(<em>a<\/em> &#8211; <em>b<\/em>)(<em>a<\/em> + <em>b<\/em>) = <em>a<\/em>^2 &#8211; <em>b<\/em>^2;$$ ainsi, les racines carr\u00e9es s&#8217;envolent&#8230; <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut ainsi conclure que :$$f<em>&#8216;<\/em>(2) = \\lim\\limits_{h\\to0} \\frac{h}{h(\\sqrt{2+h}+\\sqrt2)} =\\frac{1}{2\\sqrt2}$$ que l&#8217;on obtient en rempla\u00e7ant <em>h<\/em> par 0 dans l&#8217;expression du taux d&#8217;accroissement simplifi\u00e9e.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calcul_dune_limite_en_Terminale\"><\/span>Calcul d&#8217;une limite en Terminale<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re la fonction \\(g(x)=\\frac{\\cos x-1}{x}\\). On cherche sa limite quand <em>x<\/em> tend vers  0. Si on remplace <em>x<\/em> par 0 dans l&#8217;expression de <em>g<\/em>, on arrive \u00e0 une ind\u00e9termination du type &#8220;\\(\\frac{0}{0}\\)&#8221;. Tiens&#8230; \u00e7a nous rappelle quelque chose non ? <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On va alors poser <em>f<\/em>(<em>x<\/em>) = cos(<em>x<\/em>), dont la d\u00e9riv\u00e9e est <em>f&#8217;<\/em>(<em>x<\/em>) = -sin(<em>x<\/em>). On sait par d\u00e9finition que:$$\\lim\\limits_{x\\to0}\\frac{f(x)-f(0)}{x-0}=f'(0)$$donc: $$ \\lim\\limits_{x\\to0} g(x) =\\lim\\limits_{x\\to0}\\frac{\\cos(x)-1}{x}=-\\sin(0)=0.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Regardez cette animation : N&#8217;est-elle pas belle ?<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[145,27,29,146],"class_list":["post-1722","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-derive","tag-geogebra","tag-limite","tag-tangente"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Introduction au nombre d\u00e9riv\u00e9 - Mathweb.fr<\/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:\/\/www.mathweb.fr\/euclide\/2019\/11\/19\/introduction-au-nombre-derive\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Introduction au nombre d\u00e9riv\u00e9 - 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