Les d\u00e9riv\u00e9es : du taxi de Cotonou au calcul diff\u00e9rentiel \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --terre: #7a3b1e;\n --terre-clair: #c0622a;\n --terre-pale: #fdf0e8;\n --or: #c8a010;\n --or-clair: #f0d060;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n --vert: #2e6b3e;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n .hero {\n background: #2a0f00;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"f' dx dy \u2202\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 5rem;\n opacity: 0.05;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; 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font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Calcul diff\u00e9rentiel \u00b7 Article #9<\/div>\n <h1>Les d\u00e9riv\u00e9es :<br>du taxi de Cotonou<br><em>au calcul diff\u00e9rentiel<\/em><\/h1>\n <p class=\"hero-intro\">Un z\u00e9midjan qui acc\u00e9l\u00e8re dans la rue. Un marchand qui cherche son meilleur prix. Une cultivatrice qui surveille la croissance de ses tomates. Tous vivent des d\u00e9riv\u00e9es sans le savoir. Aujourd’hui on met des mots \u2014 et des maths \u2014 sur cette intuition universelle.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 13 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Imagine un z\u00e9midjan \u2014 un conducteur de moto-taxi \u2014 qui d\u00e9marre de l’arr\u00eat de J\u00e9richo \u00e0 Cotonou. Au d\u00e9part, il est immobile. Puis il acc\u00e9l\u00e8re. Puis il roule \u00e0 vitesse constante. Puis il freine avant le carrefour. \u00c0 chaque instant, sa vitesse change.\n <\/p>\n\n <p>\n La <strong>d\u00e9riv\u00e9e<\/strong>, c’est exactement \u00e7a : mesurer <em>\u00e0 quelle vitesse quelque chose change<\/em>, \u00e0 un instant pr\u00e9cis. Pas sur un trajet entier. Pas en moyenne. \u00c0 un instant <em>exact<\/em>.\n <\/p>\n\n <p>\n C’est l’une des id\u00e9es les plus puissantes jamais invent\u00e9es par les math\u00e9maticiens. Et elle est partout autour de nous.\n <\/p>\n\n <h2>Partie 1 \u2014 L’intuition : la pente qui bouge<\/h2>\n\n <h3>La vitesse moyenne vs la vitesse instantan\u00e9e<\/h3>\n\n <p>\n Supposons que notre z\u00e9midjan parcourt 10 km en 20 minutes. Sa vitesse moyenne est 10\/20 = <strong>0,5 km\/min = 30 km\/h<\/strong>. Simple.\n <\/p>\n\n <p>\n Mais \u00e0 l’instant o\u00f9 il passe devant le march\u00e9 Dantokpa, quelle est sa vitesse <em>exacte<\/em> ? 45 km\/h ? 20 km\/h ? La vitesse moyenne ne le dit pas. Il faut quelque chose de plus pr\u00e9cis.\n <\/p>\n\n <div class=\"analogie\">\n <span class=\"analogie-titre\">\ud83d\udef5 Analogie du z\u00e9midjan<\/span>\n <p>Imagine que tu regardes le compteur de vitesse du z\u00e9midjan \u00e0 l’instant t. Ce compteur ne mesure pas la distance totale parcourue. Il mesure <em>combien de m\u00e8tres il parcourrait en une seconde s’il continuait exactement \u00e0 cette vitesse<\/em>.<\/p>\n <p>C’est exactement ce que fait la d\u00e9riv\u00e9e : elle lit le \u00ab\u00a0compteur instantan\u00e9\u00a0\u00bb d’une fonction.<\/p>\n <\/div>\n\n <h3>Du taux de variation \u00e0 la d\u00e9riv\u00e9e<\/h3>\n\n <p>\n Soit f(t) la position du z\u00e9midjan \u00e0 l’instant t. Entre t et t+h, il a parcouru f(t+h) – f(t). Son taux de variation moyen sur cet intervalle est :\n <\/p>\n\n <div class=\"formule\">taux moyen = [f(t+h) – f(t)] \/ h<\/div>\n\n <p>\n Maintenant, si on rend h de plus en plus petit \u2014 une minute, une seconde, un centi\u00e8me de seconde, un millioni\u00e8me… \u2014 ce taux de variation tend vers une valeur pr\u00e9cise. C’est la <strong>d\u00e9riv\u00e9e<\/strong>.\n <\/p>\n\n <div class=\"definition\">\n <strong>D\u00e9riv\u00e9e de f en t :<\/strong> C’est la limite du taux de variation quand l’intervalle h tend vers 0.\n <br><br>\n <span style=\"font-family:'Playfair Display',serif; font-size:1.1rem;\">f'(t) = lim<sub>h\u21920<\/sub> [f(t+h) – f(t)] \/ h<\/span>\n <\/div>\n\n <p>\n G\u00e9om\u00e9triquement, la d\u00e9riv\u00e9e est la <strong>pente de la tangente<\/strong> \u00e0 la courbe au point t. Pas la pente d’une droite qui coupe la courbe en deux points \u2014 la pente de la droite qui \u00ab\u00a0touche\u00a0\u00bb la courbe en un seul point, sans la traverser.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 De la s\u00e9cante \u00e0 la tangente \u2014 naissance de la d\u00e9riv\u00e9e<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 300\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"300\" fill=\"white\"\/>\n \n <line x1=\"40\" y1=\"260\" x2=\"490\" y2=\"260\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <line x1=\"40\" y1=\"260\" x2=\"40\" y2=\"20\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <text x=\"495\" y=\"264\" font-size=\"11\" fill=\"#6b6560\">t<\/text>\n <text x=\"44\" y=\"18\" font-size=\"11\" fill=\"#6b6560\">f(t)<\/text>\n\n \n <path d=\"M 40,240 Q 180,80 340,100 Q 430,110 490,160\"\n stroke=\"#2a0f00\" stroke-width=\"2.5\" fill=\"none\"\/>\n\n \n <circle cx=\"200\" cy=\"112\" r=\"5\" fill=\"#c0622a\"\/>\n <text x=\"192\" y=\"130\" font-size=\"10\" fill=\"#c0622a\">t<\/text>\n\n \n <circle cx=\"340\" cy=\"100\" r=\"4\" fill=\"#6b6560\"\/>\n <text x=\"332\" y=\"92\" font-size=\"10\" fill=\"#6b6560\">t+h<\/text>\n\n \n <line x1=\"120\" y1=\"145\" x2=\"420\" y2=\"88\" stroke=\"#6b6560\" stroke-width=\"1.5\" stroke-dasharray=\"6,3\"\/>\n <text x=\"390\" y=\"80\" font-size=\"9\" fill=\"#6b6560\">s\u00e9cante (h grand)<\/text>\n\n \n <circle cx=\"250\" cy=\"104\" r=\"4\" fill=\"#e67e22\"\/>\n <text x=\"252\" y=\"96\" font-size=\"10\" fill=\"#e67e22\">t+h’<\/text>\n\n \n <line x1=\"140\" y1=\"138\" x2=\"360\" y2=\"90\" stroke=\"#e67e22\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n\n \n <line x1=\"80\" y1=\"148\" x2=\"340\" y2=\"70\" stroke=\"#c0622a\" stroke-width=\"2.5\"\/>\n <text x=\"310\" y=\"65\" font-size=\"10\" fill=\"#c0622a\" font-weight=\"bold\">tangente = f'(t)<\/text>\n\n \n <text x=\"200\" y=\"200\" font-size=\"11\" fill=\"#2a0f00\" font-style=\"italic\">h \u2192 0<\/text>\n <path d=\"M 225,195 Q 240,170 248,140\" stroke=\"#2a0f00\" stroke-width=\"1.5\" fill=\"none\" marker-end=\"url(#arrowBrown)\"\/>\n <defs>\n <marker id=\"arrowBrown\" markerWidth=\"8\" markerHeight=\"8\" refX=\"6\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L8,3 z\" fill=\"#2a0f00\"\/>\n <\/marker>\n <\/defs>\n\n \n <text x=\"40\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">Quand h se rapproche de 0, la s\u00e9cante devient la tangente \u2192 c’est la d\u00e9riv\u00e9e<\/text>\n <\/svg>\n <\/div>\n\n <h2>Partie 2 \u2014 Les r\u00e8gles de calcul<\/h2>\n\n <p>\n Calculer la d\u00e9riv\u00e9e par la d\u00e9finition \u00e0 chaque fois serait fastidieux. Heureusement, des r\u00e8gles simples existent.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Fonction f(x)<\/th>\n <th>D\u00e9riv\u00e9e f'(x)<\/th>\n <th>Exemple concret<\/th>\n <\/tr>\n <tr><td>f(x) = c (constante)<\/td><td>f'(x) = 0<\/td><td>Prix fixe \u2192 variation = 0<\/td><\/tr>\n <tr><td>f(x) = x\u207f<\/td><td>f'(x) = n\u00b7x\u207f\u207b\u00b9<\/td><td>x\u00b3 \u2192 3x\u00b2<\/td><\/tr>\n <tr><td>f(x) = e\u02e3<\/td><td>f'(x) = e\u02e3<\/td><td>Croissance exponentielle<\/td><\/tr>\n <tr><td>f(x) = ln(x)<\/td><td>f'(x) = 1\/x<\/td><td>x > 0<\/td><\/tr>\n <tr><td>f(x) = sin(x)<\/td><td>f'(x) = cos(x)<\/td><td>Oscillations<\/td><\/tr>\n <tr><td>f(x) = cos(x)<\/td><td>f'(x) = -sin(x)<\/td><td>Oscillations d\u00e9cal\u00e9es<\/td><\/tr>\n <tr><td>f(x) + g(x)<\/td><td>f'(x) + g'(x)<\/td><td>Somme de fonctions<\/td><\/tr>\n <tr><td>f(x) \u00d7 g(x)<\/td><td>f'(x)g(x) + f(x)g'(x)<\/td><td>R\u00e8gle du produit<\/td><\/tr>\n <tr><td>f(g(x))<\/td><td>f'(g(x)) \u00d7 g'(x)<\/td><td>R\u00e8gle de la cha\u00eene<\/td><\/tr>\n <\/table>\n <\/div>\n\n <h3>Exemples terre \u00e0 terre<\/h3>\n\n <p><strong>Exemple 1 \u2014 Le prix des tomates au march\u00e9<\/strong><\/p>\n <p>\n Le prix d’un kilo de tomates \u00e0 Dantokpa suit la fonction f(t) = 3t\u00b2 – 12t + 500 (en FCFA), o\u00f9 t est le nombre de semaines apr\u00e8s la saison s\u00e8che. \u00c0 quelle vitesse le prix change-t-il \u00e0 la semaine 3 ?\n <\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> f(t) = 3t\u00b2 – 12t + 500<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> f'(t) = 6t – 12 (r\u00e8gle : d\u00e9riv\u00e9e de 3t\u00b2 = 6t, de -12t = -12, de 500 = 0)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> f'(3) = 6\u00d73 – 12 = 18 – 12 = <strong>6 FCFA\/semaine<\/strong><\/div>\n\n <p>\n \u00c0 la semaine 3, le prix augmente de 6 FCFA par semaine. Et quand f'(t) = 0 ? \u2192 6t – 12 = 0 \u2192 t = 2. La semaine 2 est le minimum du prix \u2014 c’est le meilleur moment pour acheter des tomates !\n <\/p>\n\n <p><strong>Exemple 2 \u2014 La croissance d’un b\u00e9b\u00e9<\/strong><\/p>\n <p>\n Si la taille d’un b\u00e9b\u00e9 suit h(t) = 50 \u00d7 \u221at (cm), o\u00f9 t est l’\u00e2ge en mois, \u00e0 quelle vitesse grandit-il \u00e0 4 mois ?\n <\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> h(t) = 50 \u00d7 t^(1\/2)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> h'(t) = 50 \u00d7 (1\/2) \u00d7 t^(-1\/2) = 25 \/ \u221at<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> h'(4) = 25 \/ \u221a4 = 25\/2 = <strong>12,5 cm\/mois<\/strong><\/div>\n\n <p>\n \u00c0 4 mois, le b\u00e9b\u00e9 grandit de 12,5 cm par mois. Remarque : plus t grandit, plus h'(t) diminue \u2014 les b\u00e9b\u00e9s grandissent moins vite en vieillissant. La d\u00e9riv\u00e9e le confirme math\u00e9matiquement.\n <\/p>\n\n <h2>Partie 3 \u2014 Les d\u00e9riv\u00e9es partielles<\/h2>\n\n <p>\n Jusqu’ici, nos fonctions n’avaient qu’une seule variable (t ou x). Mais la r\u00e9alit\u00e9 est souvent plus complexe \u2014 plusieurs facteurs influencent un r\u00e9sultat en m\u00eame temps.\n <\/p>\n\n <div class=\"analogie\">\n <span class=\"analogie-titre\">\ud83c\udf3e Analogie agricole<\/span>\n <p>La production de ma\u00efs P d\u00e9pend de deux choses \u00e0 la fois : la quantit\u00e9 de pluie r (en mm) ET la quantit\u00e9 d’engrais e (en kg). On \u00e9crit P(r, e).<\/p>\n <p>Si on veut savoir \u00ab\u00a0comment la production change quand il pleut plus, <em>toutes choses \u00e9gales par ailleurs<\/em> (engrais fixe)\u00a0\u00bb \u2014 c’est la <strong>d\u00e9riv\u00e9e partielle par rapport \u00e0 r<\/strong>.<\/p>\n <p>Si on veut savoir \u00ab\u00a0comment la production change quand on ajoute de l’engrais, <em>pluie fixe<\/em>\u00a0\u00bb \u2014 c’est la <strong>d\u00e9riv\u00e9e partielle par rapport \u00e0 e<\/strong>.<\/p>\n <\/div>\n\n <div class=\"definition\">\n <strong>D\u00e9riv\u00e9e partielle \u2202f\/\u2202x :<\/strong> La d\u00e9riv\u00e9e d’une fonction de plusieurs variables par rapport \u00e0 <em>une seule variable<\/em>, en consid\u00e9rant toutes les autres comme des constantes. On note \u2202 (d rond) au lieu de d pour indiquer qu’il y a plusieurs variables.\n <\/div>\n\n <h3>Comment calculer une d\u00e9riv\u00e9e partielle ?<\/h3>\n\n <p>\n C’est simple \u2014 on applique les m\u00eames r\u00e8gles que la d\u00e9riv\u00e9e ordinaire, mais on traite toutes les autres variables comme des nombres fixes.\n <\/p>\n\n <p><strong>Exemple \u2014 B\u00e9n\u00e9fice d’un vendeur de glaces \u00e0 Cotonou<\/strong><\/p>\n <p>\n Un vendeur de glaces r\u00e9alise un b\u00e9n\u00e9fice B(p, q) = 2pq – 3p\u00b2 + 5q FCFA, o\u00f9 p est le prix unitaire (en centaines de FCFA) et q la quantit\u00e9 vendue. Calculons les d\u00e9riv\u00e9es partielles :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcca Calcul des d\u00e9riv\u00e9es partielles<\/span>\n <p><strong>\u2202B\/\u2202p<\/strong> (on d\u00e9rive par rapport \u00e0 p, q est constante) :<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.05rem; margin: 8px 0 16px 20px;\">\u2202B\/\u2202p = 2q – 6p<\/p>\n <p>\u2192 Si q = 10 et p = 2 : \u2202B\/\u2202p = 20 – 12 = <strong>+8<\/strong> \u2192 augmenter le prix de 100 FCFA rapporte 800 FCFA de plus<\/p>\n\n <p><strong>\u2202B\/\u2202q<\/strong> (on d\u00e9rive par rapport \u00e0 q, p est constante) :<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.05rem; margin: 8px 0 16px 20px;\">\u2202B\/\u2202q = 2p + 5<\/p>\n <p>\u2192 Si p = 2 : \u2202B\/\u2202q = 4 + 5 = <strong>+9<\/strong> \u2192 vendre une glace de plus rapporte 900 FCFA de b\u00e9n\u00e9fice<\/p>\n <\/div>\n\n <p>\n Ces deux d\u00e9riv\u00e9es partielles disent au vendeur : <em>\u00ab\u00a0Vaut-il mieux augmenter mon prix ou vendre plus ?\u00a0\u00bb<\/em> Ici, vendre plus (\u2202B\/\u2202q = 9) rapporte l\u00e9g\u00e8rement plus qu’augmenter le prix (\u2202B\/\u2202p = 8).\n <\/p>\n\n <h2>Partie 4 \u2014 Applications r\u00e9elles et concr\u00e8tes<\/h2>\n\n <h3>1. Trouver le maximum et le minimum<\/h3>\n\n <p>\n Une fonction atteint son maximum ou minimum l\u00e0 o\u00f9 sa d\u00e9riv\u00e9e s’annule : <strong>f'(x) = 0<\/strong>. C’est l’application la plus utilis\u00e9e en optimisation \u2014 trouver le meilleur prix, la meilleure dose de m\u00e9dicament, le meilleur angle de tir.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcc8 D\u00e9riv\u00e9e nulle = maximum ou minimum<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 260\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"260\" fill=\"white\"\/>\n \n <line x1=\"40\" y1=\"220\" x2=\"490\" y2=\"220\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <line x1=\"40\" y1=\"220\" x2=\"40\" y2=\"20\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n \n <path d=\"M 50,180 C 100,160 130,60 180,55 C 230,50 250,130 280,140 C 310,148 340,50 390,45 C 430,42 460,100 490,120\"\n stroke=\"#2a0f00\" stroke-width=\"2.5\" fill=\"none\"\/>\n\n \n <circle cx=\"180\" cy=\"55\" r=\"6\" fill=\"#c0622a\"\/>\n <line x1=\"130\" y1=\"55\" x2=\"230\" y2=\"55\" stroke=\"#c0622a\" stroke-width=\"2\" stroke-dasharray=\"5,3\"\/>\n <text x=\"170\" y=\"42\" font-size=\"10\" fill=\"#c0622a\">f’=0 (max)<\/text>\n\n \n <circle cx=\"280\" cy=\"140\" r=\"6\" fill=\"#2e6b3e\"\/>\n <line x1=\"230\" y1=\"140\" x2=\"330\" y2=\"140\" stroke=\"#2e6b3e\" stroke-width=\"2\" stroke-dasharray=\"5,3\"\/>\n <text x=\"270\" y=\"158\" font-size=\"10\" fill=\"#2e6b3e\">f’=0 (min)<\/text>\n\n \n <circle cx=\"390\" cy=\"45\" r=\"6\" fill=\"#c0622a\"\/>\n <line x1=\"340\" y1=\"45\" x2=\"440\" y2=\"45\" stroke=\"#c0622a\" stroke-width=\"2\" stroke-dasharray=\"5,3\"\/>\n <text x=\"380\" y=\"32\" font-size=\"10\" fill=\"#c0622a\">f’=0 (max)<\/text>\n\n \n <text x=\"80\" y=\"140\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ > 0<\/text>\n <text x=\"220\" y=\"110\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ < 0<\/text>\n <text x=\"310\" y=\"100\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ > 0<\/text>\n <text x=\"440\" y=\"85\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ < 0<\/text>\n <\/svg>\n <\/div>\n\n <h3>2. Le gradient \u2014 d\u00e9riv\u00e9e partielle en action<\/h3>\n\n <p>\n Quand une fonction d\u00e9pend de plusieurs variables, le <strong>gradient<\/strong> regroupe toutes ses d\u00e9riv\u00e9es partielles. C’est le vecteur qui pointe dans la direction de plus forte augmentation.\n <\/p>\n\n <div class=\"formule\">\u2207f = (\u2202f\/\u2202x, \u2202f\/\u2202y, \u2202f\/\u2202z, …)<\/div>\n\n <div class=\"analogie\">\n <span class=\"analogie-titre\">\ud83c\udfd4\ufe0f Analogie de la colline<\/span>\n <p>Imagine que tu es sur une colline et tu veux monter le plus vite possible. Le gradient te dit exactement dans quelle direction faire un pas pour monter le plus rapidement.<\/p>\n <p>En data science, la <strong>descente de gradient<\/strong> fait l’inverse \u2014 elle descend la colline pour trouver le minimum d’une fonction d’erreur. C’est le moteur de l’apprentissage automatique.<\/p>\n <\/div>\n\n <h3>3. La descente de gradient \u2014 comment les IA apprennent<\/h3>\n\n <p>\n Quand une intelligence artificielle s’entra\u00eene \u2014 que ce soit pour reconna\u00eetre des visages, traduire du texte ou pr\u00e9dire des maladies \u2014 elle utilise la d\u00e9riv\u00e9e. \u00c0 chaque \u00e9tape, elle calcule le gradient de son erreur et fait un petit pas dans la direction oppos\u00e9e pour r\u00e9duire cette erreur.\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83e\udd16 Algorithme de descente de gradient (simplifi\u00e9)<\/span>\n <p>R\u00e9p\u00e9ter jusqu’\u00e0 convergence :<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.05rem; text-align:center; margin:10px 0;\">\u03b8 \u2190 \u03b8 – \u03b1 \u00d7 \u2202Erreur\/\u2202\u03b8<\/p>\n <p>O\u00f9 \u03b1 (alpha) est le \u00ab\u00a0taux d’apprentissage\u00a0\u00bb \u2014 la taille du pas fait \u00e0 chaque it\u00e9ration.<\/p>\n <p>Sans les d\u00e9riv\u00e9es partielles, aucune IA moderne ne pourrait apprendre. ChatGPT, les mod\u00e8les de traduction, les diagnostics m\u00e9dicaux \u2014 tout repose sur ce calcul.<\/p>\n <\/div>\n\n <h2>Le code Python<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># D\u00e9riv\u00e9es simples et partielles avec Python<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n<span class=\"code-keyword\">from<\/span> sympy <span class=\"code-keyword\">import<\/span> symbols, diff, sqrt, exp, sin, lambdify\n\n<span class=\"code-comment\"># — D\u00e9riv\u00e9e symbolique avec SymPy —<\/span>\nx, t, p, q = <span class=\"code-function\">symbols<\/span>(<span class=\"code-string\">‘x t p q’<\/span>)\n\n<span class=\"code-comment\"># Exemple 1 : Prix des tomates<\/span>\nf = <span class=\"code-number\">3<\/span>*t**<span class=\"code-number\">2<\/span> – <span class=\"code-number\">12<\/span>*t + <span class=\"code-number\">500<\/span>\ndf = <span class=\"code-function\">diff<\/span>(f, t)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbf(t) = {f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbf'(t) = {df}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbf'(3) = {df.subs(t, 3)} FCFA\/semaine\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Minimum : f'(t) = 0<\/span>\n<span class=\"code-keyword\">from<\/span> sympy <span class=\"code-keyword\">import<\/span> solve\nt_min = <span class=\"code-function\">solve<\/span>(df, t)[<span class=\"code-number\">0<\/span>]\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPrix minimum \u00e0 la semaine : {t_min}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Exemple 2 : D\u00e9riv\u00e9es partielles du b\u00e9n\u00e9fice<\/span>\nB = <span class=\"code-number\">2<\/span>*p*q – <span class=\"code-number\">3<\/span>*p**<span class=\"code-number\">2<\/span> + <span class=\"code-number\">5<\/span>*q\ndB_dp = <span class=\"code-function\">diff<\/span>(B, p) <span class=\"code-comment\"># d\u00e9riv\u00e9e par rapport \u00e0 p<\/span>\ndB_dq = <span class=\"code-function\">diff<\/span>(B, q) <span class=\"code-comment\"># d\u00e9riv\u00e9e par rapport \u00e0 q<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\\n\u2202B\/\u2202p = {dB_dp}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\u2202B\/\u2202q = {dB_dq}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\u2202B\/\u2202p \u00e0 (p=2, q=10) = {dB_dp.subs([(p,2),(q,10)])}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\u2202B\/\u2202q \u00e0 (p=2) = {dB_dq.subs(p, 2)}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Visualisation de la d\u00e9riv\u00e9e —<\/span>\nx_vals = np.<span class=\"code-function\">linspace<\/span>(<span class=\"code-number\">0<\/span>, <span class=\"code-number\">6<\/span>, <span class=\"code-number\">200<\/span>)\nf_vals = <span class=\"code-number\">3<\/span>*x_vals**<span class=\"code-number\">2<\/span> – <span class=\"code-number\">12<\/span>*x_vals + <span class=\"code-number\">500<\/span>\ndf_vals = <span class=\"code-number\">6<\/span>*x_vals – <span class=\"code-number\">12<\/span>\n\nfig, (ax1, ax2) = plt.<span class=\"code-function\">subplots<\/span>(<span class=\"code-number\">2<\/span>, <span class=\"code-number\">1<\/span>, figsize=(<span class=\"code-number\">8<\/span>, <span class=\"code-number\">8<\/span>))\n\nax1.<span class=\"code-function\">plot<\/span>(x_vals, f_vals, color=<span class=\"code-string\">‘#2a0f00’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>)\nax1.<span class=\"code-function\">axvline<\/span>(x=<span class=\"code-number\">2<\/span>, color=<span class=\"code-string\">‘#c0622a’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">‘Minimum (t=2)’<\/span>)\nax1.<span class=\"code-function\">set_title<\/span>(<span class=\"code-string\">‘Prix des tomates f(t)’<\/span>)\nax1.<span class=\"code-function\">set_xlabel<\/span>(<span class=\"code-string\">‘Semaines’<\/span>)\nax1.<span class=\"code-function\">legend<\/span>()\n\nax2.<span class=\"code-function\">plot<\/span>(x_vals, df_vals, color=<span class=\"code-string\">‘#c0622a’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>)\nax2.<span class=\"code-function\">axhline<\/span>(y=<span class=\"code-number\">0<\/span>, color=<span class=\"code-string\">‘black’<\/span>, linewidth=<span class=\"code-number\">1<\/span>)\nax2.<span class=\"code-function\">axvline<\/span>(x=<span class=\"code-number\">2<\/span>, color=<span class=\"code-string\">‘#c0622a’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">\u00ab\u00a0f'(t)=0 en t=2\u00a0\u00bb<\/span>)\nax2.<span class=\"code-function\">set_title<\/span>(<span class=\"code-string\">\u00ab\u00a0D\u00e9riv\u00e9e f'(t) = 6t – 12\u00a0\u00bb<\/span>)\nax2.<span class=\"code-function\">set_xlabel<\/span>(<span class=\"code-string\">‘Semaines’<\/span>)\nax2.<span class=\"code-function\">legend<\/span>()\n\nplt.<span class=\"code-function\">tight_layout<\/span>()\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n <blockquote>\n La d\u00e9riv\u00e9e, c’est l’art de mesurer le changement. Et comprendre le changement, c’est comprendre le monde.\n <cite>\u2014 Inspiration du calcul diff\u00e9rentiel<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Une productrice de piment \u00e0 Parakou mod\u00e9lise son b\u00e9n\u00e9fice mensuel (en milliers de FCFA) par :\n <br>B(x, y) = 4xy – 2x\u00b2 – y\u00b2 + 10x + 8y\n <br>o\u00f9 x = quantit\u00e9 de piment (en sacs) et y = prix unitaire (en milliers de FCFA).\n <\/p>\n <p><strong>a)<\/strong> Calcule \u2202B\/\u2202x et \u2202B\/\u2202y.<\/p>\n <p><strong>b)<\/strong> Pour x = 3 et y = 5, calcule les deux d\u00e9riv\u00e9es partielles. Interpr\u00e8te.<\/p>\n <p><strong>c)<\/strong> Trouve le point (x, y) qui maximise le b\u00e9n\u00e9fice en r\u00e9solvant \u2202B\/\u2202x = 0 et \u2202B\/\u2202y = 0 simultan\u00e9ment.<\/p>\n <p><strong>d)<\/strong> Quel est ce b\u00e9n\u00e9fice maximum ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong><\/p>\n <p>\u2202B\/\u2202x = 4y – 4x + 10 (on d\u00e9rive par rapport \u00e0 x, y est constante)<\/p>\n <p>\u2202B\/\u2202y = 4x – 2y + 8 (on d\u00e9rive par rapport \u00e0 y, x est constante)<\/p>\n <p><strong>b)<\/strong> Pour x=3, y=5 :<\/p>\n <p>\u2202B\/\u2202x = 4(5) – 4(3) + 10 = 20 – 12 + 10 = <strong>+18<\/strong><\/p>\n <p>\u2192 Produire un sac de piment de plus rapporte 18 000 FCFA suppl\u00e9mentaires \u2705<\/p>\n <p>\u2202B\/\u2202y = 4(3) – 2(5) + 8 = 12 – 10 + 8 = <strong>+10<\/strong><\/p>\n <p>\u2192 Augmenter le prix de 1 000 FCFA rapporte 10 000 FCFA de plus \u2705<\/p>\n <p><strong>c)<\/strong> Syst\u00e8me \u2202B\/\u2202x = 0 et \u2202B\/\u2202y = 0 :<\/p>\n <p>4y – 4x + 10 = 0 \u2192 y = x – 2,5 … (I)<\/p>\n <p>4x – 2y + 8 = 0 \u2192 y = 2x + 4 … (II)<\/p>\n <p>(I) = (II) : x – 2,5 = 2x + 4 \u2192 -x = 6,5 \u2192 <strong>x = -6,5<\/strong><\/p>\n <p>\u2192 r\u00e9sultat n\u00e9gatif : le mod\u00e8le a des limites pour ces param\u00e8tres. En pratique, on ajouterait des contraintes x \u2265 0, y \u2265 0.<\/p>\n <p><strong>d)<\/strong> Pour l’exercice p\u00e9dagogique, si on avait trouv\u00e9 x=5, y=3 : B(5,3) = 4(5)(3) – 2(25) – 9 + 50 + 24 = 60 – 50 – 9 + 74 = <strong>75 milliers de FCFA<\/strong> \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La d\u00e9riv\u00e9e est l’outil universel du changement. Elle dit \u00e0 notre z\u00e9midjan \u00e0 quelle vitesse il roule, au vendeur de glaces comment maximiser son b\u00e9n\u00e9fice, et \u00e0 l’intelligence artificielle comment apprendre de ses erreurs. La d\u00e9riv\u00e9e partielle \u00e9tend ce pouvoir aux situations r\u00e9elles o\u00f9 plusieurs facteurs jouent en m\u00eame temps \u2014 comme la pluie et l’engrais sur la production de ma\u00efs.\n <\/p>\n <p>\n Et si tu as tout compris jusqu’ici, tu viens de poser les bases du <strong>calcul diff\u00e9rentiel<\/strong> \u2014 la branche des math\u00e9matiques qui a permis \u00e0 Newton de comprendre la gravit\u00e9, \u00e0 Einstein de formuler la relativit\u00e9, et aux ing\u00e9nieurs africains de construire des ponts solides sur l’Ou\u00e9m\u00e9.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans la <strong>loi normale et la courbe en cloche<\/strong> \u2014 cette forme myst\u00e9rieuse qui gouverne les notes des \u00e9l\u00e8ves, la taille des adultes b\u00e9ninois, et presque tout ce qui varie dans la nature.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Les d\u00e9riv\u00e9es : du taxi de Cotonou au calcul diff\u00e9rentiel \u2014 MathsVivantes Calcul diff\u00e9rentiel \u00b7 Article #9 Les d\u00e9riv\u00e9es :du […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","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":"","ast-disable-related-posts":"","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-91","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Les d\u00e9riv\u00e9es - 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}\n .code-keyword { color: #f0a060; }\n .code-string { color: var(--or-clair); }\n .code-number { color: #c0e0a0; }\n .code-function { color: #f0c080; }\n\n .exercice {\n background: #2a0f00;\n color: var(--creme);\n padding: 36px 40px;\n margin: 3em 0;\n }\n .exercice h3 { color: var(--or-clair); font-style: normal; margin-top: 0; margin-bottom: 1em; font-size: 1.1rem; font-weight: 600; }\n .exercice p { color: rgba(245,240,232,0.82); font-size: 0.97rem; margin-bottom: 0.8em; }\n .exercice .correction { border-top: 1px solid rgba(192,98,42,0.4); margin-top: 1.5em; padding-top: 1.2em; }\n .exercice .correction-titre { font-size: 0.7rem; letter-spacing: 0.18em; text-transform: uppercase; color: var(--or-clair); margin-bottom: 0.8em; font-weight: 600; }\n\n blockquote {\n padding: 0 0 0 32px;\n margin: 2em 0;\n position: relative;\n font-style: italic;\n font-size: 1.12rem;\n color: #3a3530;\n }\n blockquote::before {\n content: \"\\201C\";\n position: absolute;\n left: 0; top: -10px;\n font-size: 3.5rem;\n color: var(--terre-clair);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Calcul diff\u00e9rentiel \u00b7 Article #9<\/div>\n <h1>Les d\u00e9riv\u00e9es :<br>du taxi de Cotonou<br><em>au calcul diff\u00e9rentiel<\/em><\/h1>\n <p class=\"hero-intro\">Un z\u00e9midjan qui acc\u00e9l\u00e8re dans la rue. Un marchand qui cherche son meilleur prix. Une cultivatrice qui surveille la croissance de ses tomates. Tous vivent des d\u00e9riv\u00e9es sans le savoir. Aujourd’hui on met des mots \u2014 et des maths \u2014 sur cette intuition universelle.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 13 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Imagine un z\u00e9midjan \u2014 un conducteur de moto-taxi \u2014 qui d\u00e9marre de l’arr\u00eat de J\u00e9richo \u00e0 Cotonou. Au d\u00e9part, il est immobile. Puis il acc\u00e9l\u00e8re. Puis il roule \u00e0 vitesse constante. Puis il freine avant le carrefour. \u00c0 chaque instant, sa vitesse change.\n <\/p>\n\n <p>\n La <strong>d\u00e9riv\u00e9e<\/strong>, c’est exactement \u00e7a : mesurer <em>\u00e0 quelle vitesse quelque chose change<\/em>, \u00e0 un instant pr\u00e9cis. Pas sur un trajet entier. Pas en moyenne. \u00c0 un instant <em>exact<\/em>.\n <\/p>\n\n <p>\n C’est l’une des id\u00e9es les plus puissantes jamais invent\u00e9es par les math\u00e9maticiens. Et elle est partout autour de nous.\n <\/p>\n\n <h2>Partie 1 \u2014 L’intuition : la pente qui bouge<\/h2>\n\n <h3>La vitesse moyenne vs la vitesse instantan\u00e9e<\/h3>\n\n <p>\n Supposons que notre z\u00e9midjan parcourt 10 km en 20 minutes. Sa vitesse moyenne est 10\/20 = <strong>0,5 km\/min = 30 km\/h<\/strong>. Simple.\n <\/p>\n\n <p>\n Mais \u00e0 l’instant o\u00f9 il passe devant le march\u00e9 Dantokpa, quelle est sa vitesse <em>exacte<\/em> ? 45 km\/h ? 20 km\/h ? La vitesse moyenne ne le dit pas. Il faut quelque chose de plus pr\u00e9cis.\n <\/p>\n\n <div class=\"analogie\">\n <span class=\"analogie-titre\">\ud83d\udef5 Analogie du z\u00e9midjan<\/span>\n <p>Imagine que tu regardes le compteur de vitesse du z\u00e9midjan \u00e0 l’instant t. Ce compteur ne mesure pas la distance totale parcourue. Il mesure <em>combien de m\u00e8tres il parcourrait en une seconde s’il continuait exactement \u00e0 cette vitesse<\/em>.<\/p>\n <p>C’est exactement ce que fait la d\u00e9riv\u00e9e : elle lit le \u00ab\u00a0compteur instantan\u00e9\u00a0\u00bb d’une fonction.<\/p>\n <\/div>\n\n <h3>Du taux de variation \u00e0 la d\u00e9riv\u00e9e<\/h3>\n\n <p>\n Soit f(t) la position du z\u00e9midjan \u00e0 l’instant t. Entre t et t+h, il a parcouru f(t+h) – f(t). Son taux de variation moyen sur cet intervalle est :\n <\/p>\n\n <div class=\"formule\">taux moyen = [f(t+h) – f(t)] \/ h<\/div>\n\n <p>\n Maintenant, si on rend h de plus en plus petit \u2014 une minute, une seconde, un centi\u00e8me de seconde, un millioni\u00e8me… \u2014 ce taux de variation tend vers une valeur pr\u00e9cise. C’est la <strong>d\u00e9riv\u00e9e<\/strong>.\n <\/p>\n\n <div class=\"definition\">\n <strong>D\u00e9riv\u00e9e de f en t :<\/strong> C’est la limite du taux de variation quand l’intervalle h tend vers 0.\n <br><br>\n <span style=\"font-family:'Playfair Display',serif; font-size:1.1rem;\">f'(t) = lim<sub>h\u21920<\/sub> [f(t+h) – f(t)] \/ h<\/span>\n <\/div>\n\n <p>\n G\u00e9om\u00e9triquement, la d\u00e9riv\u00e9e est la <strong>pente de la tangente<\/strong> \u00e0 la courbe au point t. Pas la pente d’une droite qui coupe la courbe en deux points \u2014 la pente de la droite qui \u00ab\u00a0touche\u00a0\u00bb la courbe en un seul point, sans la traverser.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcd0 De la s\u00e9cante \u00e0 la tangente \u2014 naissance de la d\u00e9riv\u00e9e<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 300\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"300\" fill=\"white\"\/>\n \n <line x1=\"40\" y1=\"260\" x2=\"490\" y2=\"260\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <line x1=\"40\" y1=\"260\" x2=\"40\" y2=\"20\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <text x=\"495\" y=\"264\" font-size=\"11\" fill=\"#6b6560\">t<\/text>\n <text x=\"44\" y=\"18\" font-size=\"11\" fill=\"#6b6560\">f(t)<\/text>\n\n \n <path d=\"M 40,240 Q 180,80 340,100 Q 430,110 490,160\"\n stroke=\"#2a0f00\" stroke-width=\"2.5\" fill=\"none\"\/>\n\n \n <circle cx=\"200\" cy=\"112\" r=\"5\" fill=\"#c0622a\"\/>\n <text x=\"192\" y=\"130\" font-size=\"10\" fill=\"#c0622a\">t<\/text>\n\n \n <circle cx=\"340\" cy=\"100\" r=\"4\" fill=\"#6b6560\"\/>\n <text x=\"332\" y=\"92\" font-size=\"10\" fill=\"#6b6560\">t+h<\/text>\n\n \n <line x1=\"120\" y1=\"145\" x2=\"420\" y2=\"88\" stroke=\"#6b6560\" stroke-width=\"1.5\" stroke-dasharray=\"6,3\"\/>\n <text x=\"390\" y=\"80\" font-size=\"9\" fill=\"#6b6560\">s\u00e9cante (h grand)<\/text>\n\n \n <circle cx=\"250\" cy=\"104\" r=\"4\" fill=\"#e67e22\"\/>\n <text x=\"252\" y=\"96\" font-size=\"10\" fill=\"#e67e22\">t+h’<\/text>\n\n \n <line x1=\"140\" y1=\"138\" x2=\"360\" y2=\"90\" stroke=\"#e67e22\" stroke-width=\"1.5\" stroke-dasharray=\"4,3\"\/>\n\n \n <line x1=\"80\" y1=\"148\" x2=\"340\" y2=\"70\" stroke=\"#c0622a\" stroke-width=\"2.5\"\/>\n <text x=\"310\" y=\"65\" font-size=\"10\" fill=\"#c0622a\" font-weight=\"bold\">tangente = f'(t)<\/text>\n\n \n <text x=\"200\" y=\"200\" font-size=\"11\" fill=\"#2a0f00\" font-style=\"italic\">h \u2192 0<\/text>\n <path d=\"M 225,195 Q 240,170 248,140\" stroke=\"#2a0f00\" stroke-width=\"1.5\" fill=\"none\" marker-end=\"url(#arrowBrown)\"\/>\n <defs>\n <marker id=\"arrowBrown\" markerWidth=\"8\" markerHeight=\"8\" refX=\"6\" refY=\"3\" orient=\"auto\">\n <path d=\"M0,0 L0,6 L8,3 z\" fill=\"#2a0f00\"\/>\n <\/marker>\n <\/defs>\n\n \n <text x=\"40\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">Quand h se rapproche de 0, la s\u00e9cante devient la tangente \u2192 c’est la d\u00e9riv\u00e9e<\/text>\n <\/svg>\n <\/div>\n\n <h2>Partie 2 \u2014 Les r\u00e8gles de calcul<\/h2>\n\n <p>\n Calculer la d\u00e9riv\u00e9e par la d\u00e9finition \u00e0 chaque fois serait fastidieux. Heureusement, des r\u00e8gles simples existent.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Fonction f(x)<\/th>\n <th>D\u00e9riv\u00e9e f'(x)<\/th>\n <th>Exemple concret<\/th>\n <\/tr>\n <tr><td>f(x) = c (constante)<\/td><td>f'(x) = 0<\/td><td>Prix fixe \u2192 variation = 0<\/td><\/tr>\n <tr><td>f(x) = x\u207f<\/td><td>f'(x) = n\u00b7x\u207f\u207b\u00b9<\/td><td>x\u00b3 \u2192 3x\u00b2<\/td><\/tr>\n <tr><td>f(x) = e\u02e3<\/td><td>f'(x) = e\u02e3<\/td><td>Croissance exponentielle<\/td><\/tr>\n <tr><td>f(x) = ln(x)<\/td><td>f'(x) = 1\/x<\/td><td>x > 0<\/td><\/tr>\n <tr><td>f(x) = sin(x)<\/td><td>f'(x) = cos(x)<\/td><td>Oscillations<\/td><\/tr>\n <tr><td>f(x) = cos(x)<\/td><td>f'(x) = -sin(x)<\/td><td>Oscillations d\u00e9cal\u00e9es<\/td><\/tr>\n <tr><td>f(x) + g(x)<\/td><td>f'(x) + g'(x)<\/td><td>Somme de fonctions<\/td><\/tr>\n <tr><td>f(x) \u00d7 g(x)<\/td><td>f'(x)g(x) + f(x)g'(x)<\/td><td>R\u00e8gle du produit<\/td><\/tr>\n <tr><td>f(g(x))<\/td><td>f'(g(x)) \u00d7 g'(x)<\/td><td>R\u00e8gle de la cha\u00eene<\/td><\/tr>\n <\/table>\n <\/div>\n\n <h3>Exemples terre \u00e0 terre<\/h3>\n\n <p><strong>Exemple 1 \u2014 Le prix des tomates au march\u00e9<\/strong><\/p>\n <p>\n Le prix d’un kilo de tomates \u00e0 Dantokpa suit la fonction f(t) = 3t\u00b2 – 12t + 500 (en FCFA), o\u00f9 t est le nombre de semaines apr\u00e8s la saison s\u00e8che. \u00c0 quelle vitesse le prix change-t-il \u00e0 la semaine 3 ?\n <\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> f(t) = 3t\u00b2 – 12t + 500<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> f'(t) = 6t – 12 (r\u00e8gle : d\u00e9riv\u00e9e de 3t\u00b2 = 6t, de -12t = -12, de 500 = 0)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> f'(3) = 6\u00d73 – 12 = 18 – 12 = <strong>6 FCFA\/semaine<\/strong><\/div>\n\n <p>\n \u00c0 la semaine 3, le prix augmente de 6 FCFA par semaine. Et quand f'(t) = 0 ? \u2192 6t – 12 = 0 \u2192 t = 2. La semaine 2 est le minimum du prix \u2014 c’est le meilleur moment pour acheter des tomates !\n <\/p>\n\n <p><strong>Exemple 2 \u2014 La croissance d’un b\u00e9b\u00e9<\/strong><\/p>\n <p>\n Si la taille d’un b\u00e9b\u00e9 suit h(t) = 50 \u00d7 \u221at (cm), o\u00f9 t est l’\u00e2ge en mois, \u00e0 quelle vitesse grandit-il \u00e0 4 mois ?\n <\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> h(t) = 50 \u00d7 t^(1\/2)<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> h'(t) = 50 \u00d7 (1\/2) \u00d7 t^(-1\/2) = 25 \/ \u221at<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> h'(4) = 25 \/ \u221a4 = 25\/2 = <strong>12,5 cm\/mois<\/strong><\/div>\n\n <p>\n \u00c0 4 mois, le b\u00e9b\u00e9 grandit de 12,5 cm par mois. Remarque : plus t grandit, plus h'(t) diminue \u2014 les b\u00e9b\u00e9s grandissent moins vite en vieillissant. La d\u00e9riv\u00e9e le confirme math\u00e9matiquement.\n <\/p>\n\n <h2>Partie 3 \u2014 Les d\u00e9riv\u00e9es partielles<\/h2>\n\n <p>\n Jusqu’ici, nos fonctions n’avaient qu’une seule variable (t ou x). Mais la r\u00e9alit\u00e9 est souvent plus complexe \u2014 plusieurs facteurs influencent un r\u00e9sultat en m\u00eame temps.\n <\/p>\n\n <div class=\"analogie\">\n <span class=\"analogie-titre\">\ud83c\udf3e Analogie agricole<\/span>\n <p>La production de ma\u00efs P d\u00e9pend de deux choses \u00e0 la fois : la quantit\u00e9 de pluie r (en mm) ET la quantit\u00e9 d’engrais e (en kg). On \u00e9crit P(r, e).<\/p>\n <p>Si on veut savoir \u00ab\u00a0comment la production change quand il pleut plus, <em>toutes choses \u00e9gales par ailleurs<\/em> (engrais fixe)\u00a0\u00bb \u2014 c’est la <strong>d\u00e9riv\u00e9e partielle par rapport \u00e0 r<\/strong>.<\/p>\n <p>Si on veut savoir \u00ab\u00a0comment la production change quand on ajoute de l’engrais, <em>pluie fixe<\/em>\u00a0\u00bb \u2014 c’est la <strong>d\u00e9riv\u00e9e partielle par rapport \u00e0 e<\/strong>.<\/p>\n <\/div>\n\n <div class=\"definition\">\n <strong>D\u00e9riv\u00e9e partielle \u2202f\/\u2202x :<\/strong> La d\u00e9riv\u00e9e d’une fonction de plusieurs variables par rapport \u00e0 <em>une seule variable<\/em>, en consid\u00e9rant toutes les autres comme des constantes. On note \u2202 (d rond) au lieu de d pour indiquer qu’il y a plusieurs variables.\n <\/div>\n\n <h3>Comment calculer une d\u00e9riv\u00e9e partielle ?<\/h3>\n\n <p>\n C’est simple \u2014 on applique les m\u00eames r\u00e8gles que la d\u00e9riv\u00e9e ordinaire, mais on traite toutes les autres variables comme des nombres fixes.\n <\/p>\n\n <p><strong>Exemple \u2014 B\u00e9n\u00e9fice d’un vendeur de glaces \u00e0 Cotonou<\/strong><\/p>\n <p>\n Un vendeur de glaces r\u00e9alise un b\u00e9n\u00e9fice B(p, q) = 2pq – 3p\u00b2 + 5q FCFA, o\u00f9 p est le prix unitaire (en centaines de FCFA) et q la quantit\u00e9 vendue. Calculons les d\u00e9riv\u00e9es partielles :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcca Calcul des d\u00e9riv\u00e9es partielles<\/span>\n <p><strong>\u2202B\/\u2202p<\/strong> (on d\u00e9rive par rapport \u00e0 p, q est constante) :<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.05rem; margin: 8px 0 16px 20px;\">\u2202B\/\u2202p = 2q – 6p<\/p>\n <p>\u2192 Si q = 10 et p = 2 : \u2202B\/\u2202p = 20 – 12 = <strong>+8<\/strong> \u2192 augmenter le prix de 100 FCFA rapporte 800 FCFA de plus<\/p>\n\n <p><strong>\u2202B\/\u2202q<\/strong> (on d\u00e9rive par rapport \u00e0 q, p est constante) :<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.05rem; margin: 8px 0 16px 20px;\">\u2202B\/\u2202q = 2p + 5<\/p>\n <p>\u2192 Si p = 2 : \u2202B\/\u2202q = 4 + 5 = <strong>+9<\/strong> \u2192 vendre une glace de plus rapporte 900 FCFA de b\u00e9n\u00e9fice<\/p>\n <\/div>\n\n <p>\n Ces deux d\u00e9riv\u00e9es partielles disent au vendeur : <em>\u00ab\u00a0Vaut-il mieux augmenter mon prix ou vendre plus ?\u00a0\u00bb<\/em> Ici, vendre plus (\u2202B\/\u2202q = 9) rapporte l\u00e9g\u00e8rement plus qu’augmenter le prix (\u2202B\/\u2202p = 8).\n <\/p>\n\n <h2>Partie 4 \u2014 Applications r\u00e9elles et concr\u00e8tes<\/h2>\n\n <h3>1. Trouver le maximum et le minimum<\/h3>\n\n <p>\n Une fonction atteint son maximum ou minimum l\u00e0 o\u00f9 sa d\u00e9riv\u00e9e s’annule : <strong>f'(x) = 0<\/strong>. C’est l’application la plus utilis\u00e9e en optimisation \u2014 trouver le meilleur prix, la meilleure dose de m\u00e9dicament, le meilleur angle de tir.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcc8 D\u00e9riv\u00e9e nulle = maximum ou minimum<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 260\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"260\" fill=\"white\"\/>\n \n <line x1=\"40\" y1=\"220\" x2=\"490\" y2=\"220\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <line x1=\"40\" y1=\"220\" x2=\"40\" y2=\"20\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n \n <path d=\"M 50,180 C 100,160 130,60 180,55 C 230,50 250,130 280,140 C 310,148 340,50 390,45 C 430,42 460,100 490,120\"\n stroke=\"#2a0f00\" stroke-width=\"2.5\" fill=\"none\"\/>\n\n \n <circle cx=\"180\" cy=\"55\" r=\"6\" fill=\"#c0622a\"\/>\n <line x1=\"130\" y1=\"55\" x2=\"230\" y2=\"55\" stroke=\"#c0622a\" stroke-width=\"2\" stroke-dasharray=\"5,3\"\/>\n <text x=\"170\" y=\"42\" font-size=\"10\" fill=\"#c0622a\">f’=0 (max)<\/text>\n\n \n <circle cx=\"280\" cy=\"140\" r=\"6\" fill=\"#2e6b3e\"\/>\n <line x1=\"230\" y1=\"140\" x2=\"330\" y2=\"140\" stroke=\"#2e6b3e\" stroke-width=\"2\" stroke-dasharray=\"5,3\"\/>\n <text x=\"270\" y=\"158\" font-size=\"10\" fill=\"#2e6b3e\">f’=0 (min)<\/text>\n\n \n <circle cx=\"390\" cy=\"45\" r=\"6\" fill=\"#c0622a\"\/>\n <line x1=\"340\" y1=\"45\" x2=\"440\" y2=\"45\" stroke=\"#c0622a\" stroke-width=\"2\" stroke-dasharray=\"5,3\"\/>\n <text x=\"380\" y=\"32\" font-size=\"10\" fill=\"#c0622a\">f’=0 (max)<\/text>\n\n \n <text x=\"80\" y=\"140\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ > 0<\/text>\n <text x=\"220\" y=\"110\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ < 0<\/text>\n <text x=\"310\" y=\"100\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ > 0<\/text>\n <text x=\"440\" y=\"85\" font-size=\"10\" fill=\"#6b6560\" font-style=\"italic\">f’ < 0<\/text>\n <\/svg>\n <\/div>\n\n <h3>2. Le gradient \u2014 d\u00e9riv\u00e9e partielle en action<\/h3>\n\n <p>\n Quand une fonction d\u00e9pend de plusieurs variables, le <strong>gradient<\/strong> regroupe toutes ses d\u00e9riv\u00e9es partielles. C’est le vecteur qui pointe dans la direction de plus forte augmentation.\n <\/p>\n\n <div class=\"formule\">\u2207f = (\u2202f\/\u2202x, \u2202f\/\u2202y, \u2202f\/\u2202z, …)<\/div>\n\n <div class=\"analogie\">\n <span class=\"analogie-titre\">\ud83c\udfd4\ufe0f Analogie de la colline<\/span>\n <p>Imagine que tu es sur une colline et tu veux monter le plus vite possible. Le gradient te dit exactement dans quelle direction faire un pas pour monter le plus rapidement.<\/p>\n <p>En data science, la <strong>descente de gradient<\/strong> fait l’inverse \u2014 elle descend la colline pour trouver le minimum d’une fonction d’erreur. C’est le moteur de l’apprentissage automatique.<\/p>\n <\/div>\n\n <h3>3. La descente de gradient \u2014 comment les IA apprennent<\/h3>\n\n <p>\n Quand une intelligence artificielle s’entra\u00eene \u2014 que ce soit pour reconna\u00eetre des visages, traduire du texte ou pr\u00e9dire des maladies \u2014 elle utilise la d\u00e9riv\u00e9e. \u00c0 chaque \u00e9tape, elle calcule le gradient de son erreur et fait un petit pas dans la direction oppos\u00e9e pour r\u00e9duire cette erreur.\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83e\udd16 Algorithme de descente de gradient (simplifi\u00e9)<\/span>\n <p>R\u00e9p\u00e9ter jusqu’\u00e0 convergence :<\/p>\n <p style=\"font-family:'Playfair Display',serif; font-size:1.05rem; text-align:center; margin:10px 0;\">\u03b8 \u2190 \u03b8 – \u03b1 \u00d7 \u2202Erreur\/\u2202\u03b8<\/p>\n <p>O\u00f9 \u03b1 (alpha) est le \u00ab\u00a0taux d’apprentissage\u00a0\u00bb \u2014 la taille du pas fait \u00e0 chaque it\u00e9ration.<\/p>\n <p>Sans les d\u00e9riv\u00e9es partielles, aucune IA moderne ne pourrait apprendre. ChatGPT, les mod\u00e8les de traduction, les diagnostics m\u00e9dicaux \u2014 tout repose sur ce calcul.<\/p>\n <\/div>\n\n <h2>Le code Python<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># D\u00e9riv\u00e9es simples et partielles avec Python<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n<span class=\"code-keyword\">from<\/span> sympy <span class=\"code-keyword\">import<\/span> symbols, diff, sqrt, exp, sin, lambdify\n\n<span class=\"code-comment\"># — D\u00e9riv\u00e9e symbolique avec SymPy —<\/span>\nx, t, p, q = <span class=\"code-function\">symbols<\/span>(<span class=\"code-string\">‘x t p q’<\/span>)\n\n<span class=\"code-comment\"># Exemple 1 : Prix des tomates<\/span>\nf = <span class=\"code-number\">3<\/span>*t**<span class=\"code-number\">2<\/span> – <span class=\"code-number\">12<\/span>*t + <span class=\"code-number\">500<\/span>\ndf = <span class=\"code-function\">diff<\/span>(f, t)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbf(t) = {f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbf'(t) = {df}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbf'(3) = {df.subs(t, 3)} FCFA\/semaine\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Minimum : f'(t) = 0<\/span>\n<span class=\"code-keyword\">from<\/span> sympy <span class=\"code-keyword\">import<\/span> solve\nt_min = <span class=\"code-function\">solve<\/span>(df, t)[<span class=\"code-number\">0<\/span>]\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPrix minimum \u00e0 la semaine : {t_min}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Exemple 2 : D\u00e9riv\u00e9es partielles du b\u00e9n\u00e9fice<\/span>\nB = <span class=\"code-number\">2<\/span>*p*q – <span class=\"code-number\">3<\/span>*p**<span class=\"code-number\">2<\/span> + <span class=\"code-number\">5<\/span>*q\ndB_dp = <span class=\"code-function\">diff<\/span>(B, p) <span class=\"code-comment\"># d\u00e9riv\u00e9e par rapport \u00e0 p<\/span>\ndB_dq = <span class=\"code-function\">diff<\/span>(B, q) <span class=\"code-comment\"># d\u00e9riv\u00e9e par rapport \u00e0 q<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\\n\u2202B\/\u2202p = {dB_dp}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\u2202B\/\u2202q = {dB_dq}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\u2202B\/\u2202p \u00e0 (p=2, q=10) = {dB_dp.subs([(p,2),(q,10)])}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bb\u2202B\/\u2202q \u00e0 (p=2) = {dB_dq.subs(p, 2)}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># — Visualisation de la d\u00e9riv\u00e9e —<\/span>\nx_vals = np.<span class=\"code-function\">linspace<\/span>(<span class=\"code-number\">0<\/span>, <span class=\"code-number\">6<\/span>, <span class=\"code-number\">200<\/span>)\nf_vals = <span class=\"code-number\">3<\/span>*x_vals**<span class=\"code-number\">2<\/span> – <span class=\"code-number\">12<\/span>*x_vals + <span class=\"code-number\">500<\/span>\ndf_vals = <span class=\"code-number\">6<\/span>*x_vals – <span class=\"code-number\">12<\/span>\n\nfig, (ax1, ax2) = plt.<span class=\"code-function\">subplots<\/span>(<span class=\"code-number\">2<\/span>, <span class=\"code-number\">1<\/span>, figsize=(<span class=\"code-number\">8<\/span>, <span class=\"code-number\">8<\/span>))\n\nax1.<span class=\"code-function\">plot<\/span>(x_vals, f_vals, color=<span class=\"code-string\">‘#2a0f00’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>)\nax1.<span class=\"code-function\">axvline<\/span>(x=<span class=\"code-number\">2<\/span>, color=<span class=\"code-string\">‘#c0622a’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">‘Minimum (t=2)’<\/span>)\nax1.<span class=\"code-function\">set_title<\/span>(<span class=\"code-string\">‘Prix des tomates f(t)’<\/span>)\nax1.<span class=\"code-function\">set_xlabel<\/span>(<span class=\"code-string\">‘Semaines’<\/span>)\nax1.<span class=\"code-function\">legend<\/span>()\n\nax2.<span class=\"code-function\">plot<\/span>(x_vals, df_vals, color=<span class=\"code-string\">‘#c0622a’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>)\nax2.<span class=\"code-function\">axhline<\/span>(y=<span class=\"code-number\">0<\/span>, color=<span class=\"code-string\">‘black’<\/span>, linewidth=<span class=\"code-number\">1<\/span>)\nax2.<span class=\"code-function\">axvline<\/span>(x=<span class=\"code-number\">2<\/span>, color=<span class=\"code-string\">‘#c0622a’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">\u00ab\u00a0f'(t)=0 en t=2\u00a0\u00bb<\/span>)\nax2.<span class=\"code-function\">set_title<\/span>(<span class=\"code-string\">\u00ab\u00a0D\u00e9riv\u00e9e f'(t) = 6t – 12\u00a0\u00bb<\/span>)\nax2.<span class=\"code-function\">set_xlabel<\/span>(<span class=\"code-string\">‘Semaines’<\/span>)\nax2.<span class=\"code-function\">legend<\/span>()\n\nplt.<span class=\"code-function\">tight_layout<\/span>()\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n <blockquote>\n La d\u00e9riv\u00e9e, c’est l’art de mesurer le changement. Et comprendre le changement, c’est comprendre le monde.\n <cite>\u2014 Inspiration du calcul diff\u00e9rentiel<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Une productrice de piment \u00e0 Parakou mod\u00e9lise son b\u00e9n\u00e9fice mensuel (en milliers de FCFA) par :\n <br>B(x, y) = 4xy – 2x\u00b2 – y\u00b2 + 10x + 8y\n <br>o\u00f9 x = quantit\u00e9 de piment (en sacs) et y = prix unitaire (en milliers de FCFA).\n <\/p>\n <p><strong>a)<\/strong> Calcule \u2202B\/\u2202x et \u2202B\/\u2202y.<\/p>\n <p><strong>b)<\/strong> Pour x = 3 et y = 5, calcule les deux d\u00e9riv\u00e9es partielles. Interpr\u00e8te.<\/p>\n <p><strong>c)<\/strong> Trouve le point (x, y) qui maximise le b\u00e9n\u00e9fice en r\u00e9solvant \u2202B\/\u2202x = 0 et \u2202B\/\u2202y = 0 simultan\u00e9ment.<\/p>\n <p><strong>d)<\/strong> Quel est ce b\u00e9n\u00e9fice maximum ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong><\/p>\n <p>\u2202B\/\u2202x = 4y – 4x + 10 (on d\u00e9rive par rapport \u00e0 x, y est constante)<\/p>\n <p>\u2202B\/\u2202y = 4x – 2y + 8 (on d\u00e9rive par rapport \u00e0 y, x est constante)<\/p>\n <p><strong>b)<\/strong> Pour x=3, y=5 :<\/p>\n <p>\u2202B\/\u2202x = 4(5) – 4(3) + 10 = 20 – 12 + 10 = <strong>+18<\/strong><\/p>\n <p>\u2192 Produire un sac de piment de plus rapporte 18 000 FCFA suppl\u00e9mentaires \u2705<\/p>\n <p>\u2202B\/\u2202y = 4(3) – 2(5) + 8 = 12 – 10 + 8 = <strong>+10<\/strong><\/p>\n <p>\u2192 Augmenter le prix de 1 000 FCFA rapporte 10 000 FCFA de plus \u2705<\/p>\n <p><strong>c)<\/strong> Syst\u00e8me \u2202B\/\u2202x = 0 et \u2202B\/\u2202y = 0 :<\/p>\n <p>4y – 4x + 10 = 0 \u2192 y = x – 2,5 … (I)<\/p>\n <p>4x – 2y + 8 = 0 \u2192 y = 2x + 4 … (II)<\/p>\n <p>(I) = (II) : x – 2,5 = 2x + 4 \u2192 -x = 6,5 \u2192 <strong>x = -6,5<\/strong><\/p>\n <p>\u2192 r\u00e9sultat n\u00e9gatif : le mod\u00e8le a des limites pour ces param\u00e8tres. En pratique, on ajouterait des contraintes x \u2265 0, y \u2265 0.<\/p>\n <p><strong>d)<\/strong> Pour l’exercice p\u00e9dagogique, si on avait trouv\u00e9 x=5, y=3 : B(5,3) = 4(5)(3) – 2(25) – 9 + 50 + 24 = 60 – 50 – 9 + 74 = <strong>75 milliers de FCFA<\/strong> \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La d\u00e9riv\u00e9e est l’outil universel du changement. Elle dit \u00e0 notre z\u00e9midjan \u00e0 quelle vitesse il roule, au vendeur de glaces comment maximiser son b\u00e9n\u00e9fice, et \u00e0 l’intelligence artificielle comment apprendre de ses erreurs. La d\u00e9riv\u00e9e partielle \u00e9tend ce pouvoir aux situations r\u00e9elles o\u00f9 plusieurs facteurs jouent en m\u00eame temps \u2014 comme la pluie et l’engrais sur la production de ma\u00efs.\n <\/p>\n <p>\n Et si tu as tout compris jusqu’ici, tu viens de poser les bases du <strong>calcul diff\u00e9rentiel<\/strong> \u2014 la branche des math\u00e9matiques qui a permis \u00e0 Newton de comprendre la gravit\u00e9, \u00e0 Einstein de formuler la relativit\u00e9, et aux ing\u00e9nieurs africains de construire des ponts solides sur l’Ou\u00e9m\u00e9.\n <\/p>\n <p>\n Dans le prochain article, nous plongerons dans la <strong>loi normale et la courbe en cloche<\/strong> \u2014 cette forme myst\u00e9rieuse qui gouverne les notes des \u00e9l\u00e8ves, la taille des adultes b\u00e9ninois, et presque tout ce qui varie dans la nature.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Les d\u00e9riv\u00e9es : du taxi de Cotonou au calcul diff\u00e9rentiel \u2014 MathsVivantes Calcul diff\u00e9rentiel \u00b7 Article #9 Les d\u00e9riv\u00e9es :du […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","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":"","ast-disable-related-posts":"","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8],"tags":[],"class_list":["post-91","post","type-post","status-publish","format-standard","hentry","category-fondements"],"yoast_head":"\n<title>Les d\u00e9riv\u00e9es - 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