{"id":33377,"date":"2024-11-30T13:00:05","date_gmt":"2024-11-30T13:00:05","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33377"},"modified":"2025-07-21T03:22:46","modified_gmt":"2025-07-21T03:22:46","slug":"derivees-des-polynomes-des-fonctions-trigonometriques-et-du-logarithme","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/derivees-des-polynomes-des-fonctions-trigonometriques-et-du-logarithme\/","title":{"rendered":"D\u00e9riv\u00e9es des polyn\u00f4mes, des fonctions trigonom\u00e9triques et du logarithme"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<style><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><br \/>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>D\u00e9riv\u00e9es des polyn\u00f4mes, fonctions trigonom\u00e9triques et logarithme<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\nLa d\u00e9riv\u00e9e est un outil central du calcul diff\u00e9rentiel, avec des applications fondamentales dans les sciences, l&#8217;ing\u00e9nierie et l&#8217;\u00e9conomie. Cet article propose un guide progressif pour ma\u00eetriser la d\u00e9rivation de fonctions, allant des polyn\u00f4mes aux fonctions trigonom\u00e9triques et logarithmiques. \u00c0 travers des d\u00e9monstrations et des exemples concrets, il s&#8217;agit de comprendre \u00e0 la fois l&#8217;application des r\u00e8gles et leur fondement.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objectifs d\u2019apprentissage<\/strong><\/p>\n<ol>\n<li><strong>Comprendre<\/strong> le concept g\u00e9n\u00e9ral de d\u00e9riv\u00e9e et ses propri\u00e9t\u00e9s fondamentales.<\/li>\n<li><strong>Appliquer<\/strong> la d\u00e9finition formelle de la d\u00e9riv\u00e9e pour calculer des d\u00e9riv\u00e9es de base.<\/li>\n<li><strong>D\u00e9montrer<\/strong> \u00e0 l\u2019aide de limites la d\u00e9riv\u00e9e des fonctions constantes et de la fonction identit\u00e9.<\/li>\n<li><strong>Obtenir<\/strong> les r\u00e8gles de d\u00e9rivation des fonctions trigonom\u00e9triques \u00e0 partir des d\u00e9riv\u00e9es fondamentales du sinus et du cosinus.<\/li>\n<li><strong>Calculer<\/strong> les d\u00e9riv\u00e9es de fonctions trigonom\u00e9triques compos\u00e9es en utilisant des r\u00e8gles alg\u00e9briques.<\/li>\n<li><strong>D\u00e9montrer<\/strong> formellement la d\u00e9riv\u00e9e du logarithme naturel \u00e0 l\u2019aide de limites.<\/li>\n<\/ol>\n<p style=\"text-align:center\">\n<strong><u>TABLE DES MATI\u00c8RES<\/u> :<\/strong><br \/>\n<a href=\"#1\">D\u00e9riv\u00e9e des fonctions alg\u00e9briques<\/a><br \/>\n<a href=\"#2\">D\u00e9riv\u00e9es des fonctions transcendantes<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" title=\"YouTube video player\" data-src=\"https:\/\/www.youtube.com\/embed\/LudVXLRnmLw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>Jusqu\u2019\u00e0 pr\u00e9sent, nous avons seulement examin\u00e9 ce qu\u2019est une d\u00e9riv\u00e9e et quelques-unes de ses propri\u00e9t\u00e9s alg\u00e9briques, mais nous n\u2019avons encore rien dit sur la mani\u00e8re de les calculer. Nous allons ici r\u00e9soudre ce probl\u00e8me en montrant chacune des techniques de d\u00e9rivation et comment elles sont obtenues dans chaque cas.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>D\u00e9riv\u00e9e des fonctions alg\u00e9briques<\/h2>\n<h3>Fonction constante<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=104s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = c,<\/span> avec <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> <\/strong><\/a>une constante r\u00e9elle quelconque, alors on a :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{d}{dx}c = 0<\/span>\n<p><span style=\"color: #000080;\"><strong>D\u00c9MONSTRATION :<\/strong><\/span> En r\u00e9alit\u00e9, cette d\u00e9monstration se fait en une seule \u00e9tape :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}c &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{c - c}{\\Delta x} \\quad \\text{; D\u00e9finition de la d\u00e9riv\u00e9e de $f(x)=c$} \\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{0}{\\Delta x} = 0\n\n\\end{array}\n\n<\/span>\n<h3>Fonction identit\u00e9<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x,<\/span> alors :<\/span><\/strong><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx}{dx}=1<\/span>\n<p><span style=\"color: #000080;\"><strong>D\u00c9MONSTRATION :<\/strong><\/span> Presque identique \u00e0 la pr\u00e9c\u00e9dente, elle se fait \u00e9galement en une seule \u00e9tape :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x &amp;= \\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{(x+\\Delta x) - x}{\\Delta x} \\quad \\text{; D\u00e9finition de la d\u00e9riv\u00e9e pour $f(x) = x$}\\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\Delta x}{\\Delta x} = 1\n\n\\end{array}\n\n<\/span>\n<h3>Puissances naturelles<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=239s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x^n,<\/span> o\u00f9<\/span> <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> est un entier naturel quelconque, alors on aura :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx^n}{dx} =nx^{n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>D\u00c9MONSTRATION :<\/strong><\/span> Pour d\u00e9montrer ce th\u00e9or\u00e8me, nous devons utiliser le <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" target=\"_blank\" rel=\"noopener\">th\u00e9or\u00e8me du bin\u00f4me de Newton<\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}x^n = \\lim_{\\Delta x \\to 0} \\frac{(x+\\Delta x)^n -x^n}{\\Delta x} &amp;\\text{ ; D\u00e9finition de la limite pour $f(x)= x^n$} \\\\ \\\\\n\n&amp; \\displaystyle \\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=0}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{ ; Th\u00e9or\u00e8me du bin\u00f4me de Newton appliqu\u00e9 \u00e0 (1)} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle x^n + \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{ ; S\u00e9paration du premier terme de la somme} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right]}{\\Delta x} &amp; \\text{ ; Annulation des termes similaires} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[ {{n}\\choose{1}} x^{n-1}(\\Delta x)^{0} + \\sum_{k=2}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\text{ ; Extraction du premier terme de la somme} \\\\ \\\\\n\n&amp; \\displaystyle \\color{blue} {\\displaystyle \\dfrac{d}{dx}x^n} = n x^{n-1} &amp; \\color{black}\n\n\\end{array}\n\n<\/span>\n<h3>Puissances enti\u00e8res<\/h3>\n<p>La d\u00e9monstration pr\u00e9c\u00e9dente ne justifie que le cas o\u00f9 les puissances sont des nombres naturels, mais elle peut \u00eatre \u00e9tendue \u00e0 n&#8217;importe quel entier. Si <span class=\"katex-eq\" data-katex-display=\"false\">a\\in \\mathbb{Z}<\/span>, alors on aura :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^a}{dx} = ax^{a-1}<\/span>\n<p>Nous savons d\u00e9j\u00e0 que cela fonctionne pour les entiers positifs ; il suffit donc d&#8217;examiner ce qui se passe lorsqu\u2019on consid\u00e8re des puissances n\u00e9gatives. Il est donc suffisant de montrer que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^{-n}}{dx} = {-n}x^{-n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>D\u00c9MONSTRATION :<\/strong><\/span> Pour cela, il suffit de consid\u00e9rer la d\u00e9riv\u00e9e d\u2019un quotient :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x^{-n} &amp;= \\dfrac{d}{dx} \\left( \\dfrac{1}{x^n}\\right) \\\\ \\\\\n\n&amp; &amp;= \\dfrac{0 \\cdot nx^{n-1} - nx^{n-1} \\cdot 1}{x^{2n}}\\\\ \\\\\n\n&amp; &amp;= -nx^{n-1-2n} \\\\ \\\\\n\n&amp; &amp;= -nx^{-n-1}\n\n\\end{array}\n\n<\/span>\n<p><a name=\"2\"><\/a><\/p>\n<h2>D\u00e9riv\u00e9es des fonctions transcendantes<\/h2>\n<h3>Fonctions trigonom\u00e9triques<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=667s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Celles-ci regroupent les r\u00e8gles de d\u00e9rivation suivantes :<\/span><\/strong><\/a><\/p>\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sin(x) = \\cos(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sec(x) = \\sec(x)\\tan(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cos(x) = -\\sin(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\csc(x) = -\\csc(x)\\cot(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\tan(x) = \\sec^2(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cot(x) = -\\csc^2(x)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Pour obtenir chacune de ces r\u00e8gles, le meilleur chemin est de commencer par les d\u00e9riv\u00e9es du sinus et du cosinus ; puis, \u00e0 partir de ces r\u00e9sultats, en utilisant l\u2019alg\u00e8bre des d\u00e9riv\u00e9es, on peut obtenir celles des autres fonctions trigonom\u00e9triques.<\/p>\n<h4>D\u00e9monstration de la d\u00e9riv\u00e9e du sinus<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp;\\dfrac{d}{dx}\\sin(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x+\\Delta x) - \\sin(x)}{\\Delta x} &amp; \\text{; D\u00e9finition de la d\u00e9riv\u00e9e du sinus} \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x)\\cos(\\Delta x) + \\sin(\\Delta x)\\cos(x) - \\sin(x)}{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\sin(x)\\left[\\cos(\\Delta x) -1\\right] + \\sin(\\Delta x)\\cos(x) }{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\sin(x)\\lim_{\\Delta x \\to 0} \\left[\\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\right] + \\cos(x) \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\right] &amp; \\\\ \\\\\n\n(2)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} = 1 &amp; \\text{; Par le th\u00e9or\u00e8me du Sandwich} \\\\ \\\\\n\n(3)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\cdot \\dfrac{\\cos(\\Delta x) + 1}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos^2(\\Delta x) - 1}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{-\\sin^2(\\Delta x)}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\cdot \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- (1)\\cdot(0) = 0 \\\\ \\\\\n\n(4) &amp;\\color{blue}\\dfrac{d}{dx}\\sin(x) = \\cos(x) \\color{black} &amp; \\text{; D\u2019apr\u00e8s (1,2,3)}\n\n\\end{array}<\/span>\n<h4>D\u00e9monstration de la d\u00e9riv\u00e9e du cosinus<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}\\cos(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x + \\Delta x) - \\cos(x)}{\\Delta x} &amp; \\text{; D\u00e9finition de la d\u00e9riv\u00e9e du cosinus} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x)\\cos(\\Delta x) - \\sin(x)\\sin(\\Delta x) - \\cos(x)}{\\Delta x} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x) [ \\cos(\\Delta x) - 1] - \\sin(x)\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ [ \\cos(\\Delta x) - 1]}{\\Delta x} - \\sin(x) \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\cdot(0) - \\sin(x)\\cdot (1)\\\\ \\\\\n\n&amp;\\color{blue}\\dfrac{d}{dx}\\cos(x) = - \\sin(x) \\color{black}\n\n\\end{array}<\/span>\n<h4>D\u00e9riv\u00e9es de la tangente, s\u00e9cante, cos\u00e9cante et cotangente<\/h4>\n<p>Ayant obtenu les r\u00e9sultats pour le sinus et le cosinus, les d\u00e9riv\u00e9es des autres fonctions trigonom\u00e9triques s\u2019obtiennent d\u00e9sormais facilement.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\tan(x) &amp;= \\dfrac{d}{dx} \\left( \\dfrac{\\sin(x)}{\\cos(x)} \\right) = \\dfrac{\\cos^2(x) + \\sin^2(x)}{\\cos^2(x)} = \\dfrac{1}{\\cos^2(x)} = \\color{blue}\\sec^2(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\sec(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\cos(x)} \\right) = \\dfrac{\\sin(x)}{\\cos^2(x)} =\\color{blue}\\sec(x)\\tan(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\csc(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\sin(x)}\\right) = -\\dfrac{\\cos(x)}{\\sin^2(x)} =\\color{blue} - \\csc(x)\\cot(x)\\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx} \\cot(x) &amp;= \\dfrac{d}{dx} \\left(\\dfrac{\\cos(x)}{\\sin(x)}\\right) = \\dfrac{-\\sin^2(x)-\\cos^2(x)}{\\sin^2(x)} = -\\dfrac{1}{\\sin^2(x)} =\\color{blue} -\\csc^2(x)\\color{black}\n\n\\end{array}<\/span>\n<h3>D\u00e9riv\u00e9e des fonctions logarithmiques<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=1331s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La d\u00e9riv\u00e9e du logarithme naturel<\/span><\/strong><\/a> est donn\u00e9e par :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\ln(x) = \\frac{1}{x}<\/span>\n<p><span style=\"color: #000080;\"><strong>D\u00c9MONSTRATION :<\/strong><\/span> En raisonnant \u00e0 partir de la d\u00e9finition de la d\u00e9riv\u00e9e, on obtient le raisonnement suivant :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx} \\ln(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\left [\\dfrac{\\ln(x+\\Delta x) - \\ln(x)}{\\Delta x} \\right] &amp;\\text{; D\u00e9finition de la d\u00e9riv\u00e9e du logarithme naturel} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{\\Delta x} \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right) \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\color{red}x\\color{black}} \\frac{\\color{red}x\\color{black}}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{x} \\ln \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} =\\dfrac{1}{x} \\ln \\displaystyle \\left[ \\lim_{\\Delta x \\to 0} \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n(2) &amp; n=\\dfrac{x}{\\Delta x} &amp; \\text{; Substitution} \\\\ \\\\\n\n(3) &amp; (\\Delta x \\to 0^+) \\longrightarrow (n\\to +\\infty) \\\\ \\\\\n\n(4) &amp; \\dfrac{d}{dx} \\ln(x) = \\dfrac{1}{x} \\ln\\left[ \\displaystyle \\lim_{n \\to +\\infty} \\left(1 + \\dfrac{1}{n} \\right)^n \\right] = \\dfrac{1}{x} \\ln(e) = \\color{blue}\\dfrac{1}{x} \\color{black} &amp; \\text{; D\u2019apr\u00e8s (1,2,3)}\n\n\\end{array}<\/span>\n<p>\nNous avons ainsi parcouru, \u00e9tape par \u00e9tape, les d\u00e9riv\u00e9es fondamentales que tout \u00e9tudiant doit ma\u00eetriser : des fonctions alg\u00e9briques de base jusqu&#8217;aux principales fonctions transcendantes telles que les fonctions trigonom\u00e9triques et le logarithme naturel. En ma\u00eetrisant ces d\u00e9monstrations, vous serez en mesure d\u2019appliquer les r\u00e8gles de d\u00e9rivation, de comprendre leur origine ainsi que leur justification formelle. Cette connaissance constitue la base n\u00e9cessaire pour aborder avec confiance des probl\u00e8mes plus complexes n\u00e9cessitant une analyse pr\u00e9cise du changement.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>D\u00e9riv\u00e9es des polyn\u00f4mes, fonctions trigonom\u00e9triques et logarithme La d\u00e9riv\u00e9e est un outil central du calcul diff\u00e9rentiel, avec des applications fondamentales dans les sciences, l&#8217;ing\u00e9nierie et l&#8217;\u00e9conomie. Cet article propose un guide progressif pour ma\u00eetriser la d\u00e9rivation de fonctions, allant des polyn\u00f4mes aux fonctions trigonom\u00e9triques et logarithmiques. \u00c0 travers des d\u00e9monstrations et des exemples concrets, il [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33155,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":11,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-33377","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calcul-differentiel","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>D\u00e9riv\u00e9es des polyn\u00f4mes, des fonctions trigonom\u00e9triques et du logarithme - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez pas \u00e0 pas \u00e0 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