{"id":4007,"date":"2025-01-13T09:26:00","date_gmt":"2025-01-13T08:26:00","guid":{"rendered":"https:\/\/jhmaths.fr\/?page_id=4007"},"modified":"2025-01-26T15:32:11","modified_gmt":"2025-01-26T14:32:11","slug":"nombre-derive-et-tangente","status":"publish","type":"page","link":"https:\/\/jhmaths.fr\/index.php\/premiere-generale\/nombre-derive-et-tangente\/","title":{"rendered":"Nombre d\u00e9riv\u00e9 et tangente"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">D&rsquo;abord, un peu d&rsquo;histoire<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Comment comprendre FACILEMENT les d\u00e9riv\u00e9es\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RLEE-iSBimc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Taux de variation et Coefficient directeur<\/h2>\n\n\n\n<p class=\"has-text-align-left propriete wp-block-paragraph\">Le taux de variation d&rsquo;une fonction \\(f\\) entre \\(a\\) et \\(b\\) vaut <br>\\[\\displaystyle m=\\frac{f(b)-f(a)}{b-a}\\]<br>Le taux de variation d&rsquo;une fonction \\(f\\) en \\(a\\) vaut<br>\\[\\displaystyle m=\\frac{f(a+h)-f(a)}{h}\\]<br>(on a remplac\u00e9 \\(b\\) par \\(a+h\\) )<\/p>\n\n\n\n<p class=\"has-text-align-left propriete wp-block-paragraph\">Une droite \\((AB)\\) a pour \u00e9quation \\(f(x)=mx+p\\) ;<br>\\(m\\) est le coefficient directeur ; il vaut<br>\\[\\displaystyle m=\\frac{y_B-y_A}{x_B-x_A}\\]<br>(c&rsquo;est le taux de variation de \\(f\\) entre \\(x_A\\) et \\(x_B\\) )<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Calculer le coefficient directeur d&#039;une droite\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/AeOvx12gDtk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Nombre d\u00e9riv\u00e9 et tangente<\/h2>\n\n\n\n<p class=\"propriete wp-block-paragraph\">Si \\(f\\) est une fonction, <br>\\(\\mathscr{C}\\) sa courbe <br>et \\(A\\) est un point de la courbe qui a pour abscisse \\(a\\).<br><br>Alors le <strong>nombre d\u00e9riv\u00e9 est le coefficient directeur de la tangente \u00e0 la courbe au point \\(A\\)<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"249\" src=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/04\/tangente-300x249.png\" alt=\"\" class=\"wp-image-2842\" srcset=\"https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/04\/tangente-300x249.png 300w, https:\/\/jhmaths.fr\/wp-content\/uploads\/2020\/04\/tangente.png 550w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"d\u00e9rivation \u2022 fonction \u2022 comprendre la d\u00e9finition \u2022 cours IMPORTANT \u2022 Premi\u00e8re sp\u00e9 maths S ES STI\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/cv0djQqfwBg?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">En pratique<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Calculer le nombre d\u00e9riv\u00e9 (1) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UmT0Gov6yyE?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<div style=\"height:63px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Quizz<\/h2>\n\n\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-64\" class=\"h5p-iframe\" data-content-id=\"64\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"1STHR nombre d\u00e9riv\u00e9 (1)\"><\/iframe><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">\u00c9quation de la tangente<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"DEMONSTRATION : Equation de la tangente \u00e0 une courbe - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Jj0ql6-o2Uo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Application<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9terminer une \u00e9quation de la tangente \u00e0 une courbe - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bELc3OM9osQ?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Fonction d\u00e9riv\u00e9e<\/h2>\n\n\n\n<p class=\"has-text-align-left propriete wp-block-paragraph\">En associant \u00e0 tout nombre $x$ le nombre d\u00e9riv\u00e9 $f'(x)$, on construit une fonction appel\u00e9e fonction d\u00e9riv\u00e9e de $f$ et not\u00e9e $f'(x)$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Tableau des fonctions d\u00e9riv\u00e9es des fonctions de r\u00e9f\u00e9rence<\/h2>\n\n\n\n<figure class=\"wp-block-table propriete\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>Fonction<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>D\u00e9riv\u00e9e<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">k (constante)<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$ax+b$ (fct affine)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$a$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$x^2$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$2x$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$x^n$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$nx^{n-1}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{1}{x}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{-1}{x^2}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{1}{x^n}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{-n}{x^{n+1}}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\sqrt{x}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{1}{2\\sqrt{x}}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Tableau des op\u00e9rations sur les fonctions d\u00e9riv\u00e9es<\/h2>\n\n\n\n<figure class=\"wp-block-table propriete\"><table class=\"has-fixed-layout\"><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>$u$ et $v$ sont deux fonctions<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>leurs d\u00e9riv\u00e9es sont $u&rsquo;$ et $v&rsquo;$<\/strong><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$k \\times u$ ($k$ est une constante)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$k \\times u&rsquo;$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$u+v$ (somme)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$u&rsquo;+v&rsquo;$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$uv$ (produit)<\/td><td class=\"has-text-align-center\" data-align=\"center\">$u&rsquo;v+v&rsquo;u$ (attention ce n&rsquo;est pas intuitif ! et en plus il y a un plus !)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{u}{v}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{-v&rsquo;}{v^2}$ (pas intuitif non plus, et il y a un moins)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{u}{v}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{u&rsquo;v-v&rsquo;u}{v^2}$ (pas intuitif non plus, et il y a encore un moins)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$u^n$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$nu&rsquo;u^{n-1}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\sqrt{u}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{u&rsquo;}{2\\sqrt{u}}$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"has-text-align-left wp-block-paragraph\">Exemple : Si $f(x)=x^2(5x-4)$ alors la d\u00e9riv\u00e9e est <br>\\begin{align*}<br>f'(x) &amp; =2x(5x-4)+5 \\times x^2 &amp; \\text{ avec } &amp;u=x^2 \\text{ et } v=5x-4 \\\\<br> &amp;=10x^2-8x+5x^2 &amp; &amp;u&rsquo;=2x \\quad v&rsquo;=5\\\\<br>&amp; =15x^2-8x<br>\\end{align*}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Des exemples en vid\u00e9o<\/h3>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9river une fonction (1) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ehHoLK98Ht0?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9river une fonction (2) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1fOGueiO_zk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9river une fonction (3) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OMsZNNIIdrw?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9river une fonction (4) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jOuC7aq3YkM?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"D\u00e9river une fonction (5) - Premi\u00e8re\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-MfEczGz_6Y?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>D&rsquo;abord, un peu d&rsquo;histoire Taux de variation et Coefficient directeur Le taux de variation d&rsquo;une fonction \\(f\\) entre \\(a\\) et \\(b\\) vaut \\[\\displaystyle m=\\frac{f(b)-f(a)}{b-a}\\]Le taux de variation d&rsquo;une fonction \\(f\\) en \\(a\\) vaut\\[\\displaystyle m=\\frac{f(a+h)-f(a)}{h}\\](on a remplac\u00e9 \\(b\\) par \\(a+h\\) ) Une droite \\((AB)\\) a pour \u00e9quation \\(f(x)=mx+p\\) ;\\(m\\) est le coefficient directeur ; il vaut\\[\\displaystyle [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3972,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"folder":[],"class_list":["post-4007","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/4007","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/comments?post=4007"}],"version-history":[{"count":20,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/4007\/revisions"}],"predecessor-version":[{"id":4042,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/4007\/revisions\/4042"}],"up":[{"embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/3972"}],"wp:attachment":[{"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/media?parent=4007"}],"wp:term":[{"taxonomy":"folder","embeddable":true,"href":"https:\/\/jhmaths.fr\/index.php\/wp-json\/wp\/v2\/folder?post=4007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}