{"id":29732,"date":"2024-11-27T12:00:17","date_gmt":"2024-11-27T12:00:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29732"},"modified":"2024-11-27T17:37:27","modified_gmt":"2024-11-27T17:37:27","slug":"la-derivee-comme-limite-dune-fonction","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/","title":{"rendered":"La d\u00e9riv\u00e9e comme limite d&#8217;une fonction"},"content":{"rendered":"<style>\np {\n  text-align: justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">La d\u00e9riv\u00e9e comme limite d&#8217;une fonction<\/h1>\n<p style=\"text-align:center;\">\n  <em><strong>R\u00e9sum\u00e9 :<\/strong> Dans ce cours, nous explorerons le concept de la d\u00e9riv\u00e9e comme un outil math\u00e9matique permettant d&#8217;analyser les variations des fonctions. Nous partirons de la pente d&#8217;une ligne s\u00e9cante et, en calculant la limite lorsque les points se rapprochent, nous d\u00e9finirons la d\u00e9riv\u00e9e comme la pente de la tangente. De plus, nous \u00e9tudierons ses propri\u00e9t\u00e9s principales et ses r\u00e8gles, telles que les r\u00e8gles de somme, de produit et de quotient, qui sont fondamentales pour appliquer les d\u00e9riv\u00e9es dans l&#8217;analyse des fonctions et des ph\u00e9nom\u00e8nes de changement.<\/em>\n<\/p>\n<p style=\"text-align:center;\"><strong>Objectifs d&#8217;apprentissage<\/strong><\/p>\n<p>\u00c0 la fin de ce cours, les \u00e9tudiants seront capables de :<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> la d\u00e9riv\u00e9e comme limite d\u00e9crivant le changement instantan\u00e9 d&#8217;une fonction et comme la pente de la tangente \u00e0 une courbe en un point.<\/li>\n<li><strong>Expliquer<\/strong> comment la diff\u00e9rentiabilit\u00e9 implique la continuit\u00e9 des fonctions.<\/li>\n<li><strong>D\u00e9montrer<\/strong> les r\u00e8gles de base de la d\u00e9rivation \u00e0 partir de la d\u00e9finition formelle.<\/li>\n<li><strong>Appliquer<\/strong> les propri\u00e9t\u00e9s alg\u00e9briques des d\u00e9riv\u00e9es (somme, produit et quotient) dans des probl\u00e8mes math\u00e9matiques.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>Table des mati\u00e8res :<\/u><\/strong><\/p>\n<p><a href=\"#1\"><strong>Le concept de d\u00e9riv\u00e9e<\/strong><\/a><br \/>\n<a href=\"#1\">La pente de la ligne s\u00e9cante<\/a><br \/>\n<a href=\"#1\">Passage \u00e0 la limite : la d\u00e9riv\u00e9e et la pente de la tangente<\/a><br \/>\n<a href=\"#1\">D\u00e9finition alternative<\/a><br \/>\n<a href=\"#1\"><strong>Propri\u00e9t\u00e9s des d\u00e9riv\u00e9es<\/strong><\/a><br \/>\n<a href=\"#1\">La diff\u00e9rentiabilit\u00e9 implique la continuit\u00e9<\/a><br \/>\n<a href=\"#1\">Alg\u00e8bre des d\u00e9riv\u00e9es<\/a><\/p>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/TFxATgmYvkY\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Le concept de d\u00e9riv\u00e9e<\/h2>\n<p>La nature est g\u00e9n\u00e9ralement sujette \u00e0 des changements, et l&#8217;outil math\u00e9matique par excellence pour calculer et comprendre ces changements est la d\u00e9riv\u00e9e. Elle \u00e9merge de la question : \u00ab Que se passe-t-il avec la valeur d&#8217;une fonction <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> lorsque la variable <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> est augment\u00e9e ou diminu\u00e9e d&#8217;une quantit\u00e9 aussi petite que souhait\u00e9e <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> ? \u00bb Le concept de d\u00e9riv\u00e9e appara\u00eet comme la limite d&#8217;une fonction en analysant cette question.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>La pente de la ligne s\u00e9cante<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=164s\" target=\"_blank\" rel=\"noopener\"><strong>Consid\u00e9rons une fonction<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span>, \u00e9valu\u00e9e en deux points <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. Toute ligne passant par deux points d&#8217;une courbe est appel\u00e9e \u00ab ligne s\u00e9cante \u00bb et ressemble \u00e0 ce qui est montr\u00e9 sur la figure ci-dessous.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"Graphique de la ligne s\u00e9cante\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"Graphique de la ligne s\u00e9cante\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><\/noscript><\/p>\n<p>La pente de cette ligne s\u00e9cante est donn\u00e9e par :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Passage \u00e0 la limite : la d\u00e9riv\u00e9e et la pente de la tangente<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=278s\" target=\"_blank\" rel=\"noopener\"><strong>Consid\u00e9rons la ligne s\u00e9cante de la courbe<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span>, passant par <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. En calculant la limite lorsque <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> tend vers z\u00e9ro, nous obtenons la tangente \u00e0 la courbe passant par <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, f(x_0)).<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"Graphique de la ligne tangente\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"Graphique de la ligne tangente\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><\/noscript><\/p>\n<p>\u00c0 partir de cela, la d\u00e9finition formelle de la d\u00e9riv\u00e9e d&#8217;une fonction <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> en un point <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> est donn\u00e9e par :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{df(x_0)}{dx}:= \\lim_{\\Delta x \\to 0}\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>Cela repr\u00e9sente \u00e9galement la pente de la tangente passant par <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>D\u00e9finition alternative<\/h3>\n<p>Une autre mani\u00e8re de pr\u00e9senter la d\u00e9finition de la d\u00e9riv\u00e9e comme limite repose sur le remplacement suivant :<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nx_i &amp;= x_0\\\\\n\nx_f &amp;= x_i + \\Delta x\n\n\\end{array}\n\n<\/span>\n<p>Ainsi, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_f - x_i<\/span>, et la d\u00e9finition de la d\u00e9riv\u00e9e devient :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\dfrac{df(x_i)}{dx} &amp;=\\displaystyle \\lim_{\\Delta x \\to 0}\\dfrac{ f(x_i + \\Delta x) - f(x_i)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{x_f - x_i \\to 0} \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\\\\ \\\\\n\n&amp;=\\displaystyle  \\lim_{x_f \\to x_i } \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\n\n\\end{array}\n\n<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"D\u00e9finition de la d\u00e9riv\u00e9e comme limite des pentes des s\u00e9cantes\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"D\u00e9finition de la d\u00e9riv\u00e9e comme limite des pentes des s\u00e9cantes\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><\/noscript><\/p>\n<p>Les deux d\u00e9finitions sont \u00e9quivalentes et peuvent \u00eatre utilis\u00e9es de mani\u00e8re interchangeable selon les besoins.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Propri\u00e9t\u00e9s des d\u00e9riv\u00e9es<\/h2>\n<p>Une fonction est dite d\u00e9rivable en <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> si la limite suivante existe :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>Nous disons qu&#8217;une fonction est d\u00e9rivable sur un ensemble <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> si cette limite est bien d\u00e9finie pour tous les <span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I.<\/span> Les fonctions d\u00e9rivables poss\u00e8dent les propri\u00e9t\u00e9s suivantes :<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>La diff\u00e9rentiabilit\u00e9 implique la continuit\u00e9<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=526s\" target=\"_blank\" rel=\"noopener\"><strong>Si une fonction est d\u00e9rivable en<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> alors elle est continue en <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span> Cela peut \u00eatre d\u00e9montr\u00e9 par le raisonnement suivant :<\/p>\n<p>Pour que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> soit continue en <span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> il est n\u00e9cessaire que :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = f(x_0)<\/span>\n<p>En examinant le c\u00f4t\u00e9 gauche de cette expression, nous avons :<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to x_0} f(x) &amp;= \\displaystyle \\lim_{x\\to x_0} \\left[ f(x) + f(x_0) - f(x_0) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( f(x)  - f(x_0) \\right) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0)  \\right] \\\\ \\\\\n\n&amp;=f(x_0) +\\displaystyle \\lim_{x\\to x_0} \\left[ \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0) \\right]\n\\end{array}\n\n<\/span>\n<p>Par cons\u00e9quent, pour que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> soit continue en <span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> la limite du c\u00f4t\u00e9 droit doit \u00eatre bien d\u00e9finie. Cela se produit si et seulement si :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\dfrac{f(x) - f(x_0)}{x-x_0} = \\dfrac{df(x_0)}{dx}<\/span>\n<p>En d&#8217;autres termes, si <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> est d\u00e9rivable en <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span> Par cons\u00e9quent, la diff\u00e9rentiabilit\u00e9 implique la continuit\u00e9.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Alg\u00e8bre des d\u00e9riv\u00e9es<\/h3>\n<p>Soient <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> des fonctions d\u00e9rivables pour tout <span class=\"katex-eq\" data-katex-display=\"false\">x \\in I,<\/span> et soient <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta \\in \\mathbb{R}.<\/span> Alors, les propri\u00e9t\u00e9s suivantes sont v\u00e9rifi\u00e9es :<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha \\dfrac{df(x)}{dx} \\pm \\beta\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( f(x) g(x) \\right) = \\dfrac{df(x)}{dx} g(x) + f(x)\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0<\/span>, alors <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\dfrac{f(x)}{g(x)} \\right) = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx} }{\\left[g(x)\\right]^2}<\/span><\/li>\n<\/ol>\n<p>Comme nous pouvons le voir, l&#8217;alg\u00e8bre des d\u00e9riv\u00e9es peut sembler moins intuitive qu&#8217;elle ne para\u00eet au premier abord. Cependant, ces propri\u00e9t\u00e9s peuvent \u00eatre d\u00e9riv\u00e9es sans grande difficult\u00e9 \u00e0 partir de la d\u00e9finition des d\u00e9riv\u00e9es comme limites.<\/p>\n<p><span style=\"color: #000080;\">Preuve :<\/span><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=925s\" target=\"_blank\" rel=\"noopener\"><strong>La preuve de la r\u00e8gle de la somme<\/strong><\/a> est la suivante :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\alpha f(x) \\pm \\beta g(x) \\right) &amp; =\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\left[\\alpha f(x+\\Delta x) \\pm \\beta g(x+ \\Delta x)\\right] - \\left[\\alpha f(x) \\pm \\beta g(x) \\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\left[\\alpha f(x+\\Delta x) - \\alpha f(x)\\right] \\pm \\left[\\beta g(x+\\Delta x) - \\beta g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\alpha \\left[ f(x+\\Delta x) -  f(x)\\right] \\pm  \\beta  \\left[ g(x+\\Delta x) - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\alpha \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) -  f(x)}{\\Delta x} \\pm \\beta \\lim_{\\Delta x \\to 0} \\dfrac{ g(x+\\Delta x) -  g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\alpha \\dfrac{df(x)}{dx} \\pm \\beta \\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=1059s\" target=\"_blank\" rel=\"noopener\"><strong>La preuve de la r\u00e8gle du produit<\/strong><\/a> est un peu plus complexe :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left[f(x)g(x)\\right] &amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) -  f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) + \\color{red}f(x)g(x+\\Delta x) - f(x)g(x+\\Delta x) \\color{black} - f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\left[f(x+\\Delta x) - f(x) \\right] g(x+\\Delta x) + f(x) \\left[g(x+\\Delta x)  - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x) \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x)\\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;= g(x) \\dfrac{df(x)}{dx} + f(x)\\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p>Nous avons ici utilis\u00e9 le fait que, puisque <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> est une fonction d\u00e9rivable, elle est aussi continue. Par cons\u00e9quent, <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{\\Delta x\\to 0 } g(x+\\Delta x) = g(x).<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=162s\" target=\"_blank\" rel=\"noopener\"><strong>Enfin, la preuve de la r\u00e8gle du quotient<\/strong><\/a> peut \u00eatre obtenue en utilisant la r\u00e8gle du produit. Supposons une fonction de la forme <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = f(x)\/g(x),<\/span> o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0.<\/span> Alors :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{df(x)}{dx}= \\dfrac{d}{dx}(k(x)g(x)) = \\dfrac{dk(x)}{dx}g(x) + k(x)\\dfrac{dg(x)}{dx}<\/span>\n<p>En r\u00e9solvant pour <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx},<\/span> nous obtenons :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}g(x) = \\dfrac{df(x)}{dx} - k(x)\\dfrac{dg(x)}{dx} = \\dfrac{d}{dx}f(x) - \\dfrac{f(x)}{g(x)}\\dfrac{dg(x)}{dx} <\/span>\n<p>Par cons\u00e9quent :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)\n\n &amp;= \\dfrac{dk(x)}{dx} =\\dfrac{1}{g(x)} \\dfrac{df(x)}{dx} - \\dfrac{f(x)}{\\left[g(x)\\right]^2}\\dfrac{dg(x)}{dx} \\\\ \\\\\n\n&amp; = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx}}{[g(x)]^2}\n\n\\end{array}\n\n<\/span>\n<p>C&#8217;est ce que nous voulions d\u00e9montrer.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La d\u00e9riv\u00e9e comme limite d&#8217;une fonction R\u00e9sum\u00e9 : Dans ce cours, nous explorerons le concept de la d\u00e9riv\u00e9e comme un outil math\u00e9matique permettant d&#8217;analyser les variations des fonctions. Nous partirons de la pente d&#8217;une ligne s\u00e9cante et, en calculant la limite lorsque les points se rapprochent, nous d\u00e9finirons la d\u00e9riv\u00e9e comme la pente de la [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29706,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-29732","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>La d\u00e9riv\u00e9e comme limite d&#039;une fonction - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"J&#039;explique le concept de d\u00e9riv\u00e9e en tant que limite d&#039;une fonction et je vous montre de mani\u00e8re claire et simple comment en d\u00e9duire toutes ses propri\u00e9t\u00e9s.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La d\u00e9riv\u00e9e comme limite d&#039;une fonction\" \/>\n<meta property=\"og:description\" content=\"J&#039;explique le concept de d\u00e9riv\u00e9e en tant que limite d&#039;une fonction et je vous montre de mani\u00e8re claire et simple comment en d\u00e9duire toutes ses propri\u00e9t\u00e9s.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-11-27T12:00:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-11-27T17:37:27+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/derivada014-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"La d\u00e9riv\u00e9e comme limite d&#039;une fonction\" \/>\n<meta name=\"twitter:description\" content=\"J&#039;explique le concept de d\u00e9riv\u00e9e en tant que limite d&#039;une fonction et je vous montre de mani\u00e8re claire et simple comment en d\u00e9duire toutes ses propri\u00e9t\u00e9s.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/derivada014.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"La d\u00e9riv\u00e9e comme limite d&#8217;une fonction\",\"datePublished\":\"2024-11-27T12:00:17+00:00\",\"dateModified\":\"2024-11-27T17:37:27+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/\"},\"wordCount\":1800,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/derivada014.jpg\",\"articleSection\":[\"Calcul Diff\u00e9rentiel\",\"Math\u00e9matiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/\",\"url\":\"http:\/\/toposuranos.com\/material\/fr\/la-derivee-comme-limite-dune-fonction\/\",\"name\":\"La d\u00e9riv\u00e9e comme limite d'une fonction - 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