{"id":235481,"date":"2022-06-22T17:16:11","date_gmt":"2022-06-22T15:16:11","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=235481"},"modified":"2025-09-29T11:19:29","modified_gmt":"2025-09-29T09:19:29","slug":"calculer-la-derivee-dune-fonction","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/calculer-la-derivee-dune-fonction\/","title":{"rendered":"Comment calculer la d\u00e9riv\u00e9e d’une fonction ?"},"content":{"rendered":"\n

Vous cherchez \u00e0 calculer la d\u00e9riv\u00e9e d’une fonction<\/strong> ? Avec ce cours sp\u00e9cialement con\u00e7u pour cette notion, vous allez pouvoir la ma\u00eetriser pleinement gr\u00e2ce \u00e0 des m\u00e9thodologies abouties !<\/p>\n

Et pour approfondir encore plus ta compr\u00e9hension, d\u00e9couvre comment les d\u00e9riv\u00e9es fa\u00e7onnent les concepts avanc\u00e9s en pr\u00e9pa scientifique avec un cours particuliers de maths<\/strong>\u00a0<\/a>taill\u00e9 sur-mesure. \ud83d\udd0d<\/p>\n

\u00a0<\/p>\n\n\n\n

Calculs de d\u00e9riv\u00e9es <\/h2>\n\n\n\n

Th\u00e9or\u00e8me : Calculer la d\u00e9riv\u00e9e d’une fonction <\/h3>\n\n\n\n

<\/p>\nSoient \"f:I\to \mathbb{R}\" et \"g:I\to\mathbb{R}\" deux fonctions et \"a\in I\". On suppose que \"f\" et \"g\" sont d\u00e9rivables en \"a\".
\nAlors,\n

  • pour tout \"(\lambda,\mu)\in\mathbb{R}^2\", la fonction \"\lambda.f+\mu.g\" est d\u00e9rivable en \"a\" et :\n

      <\/span>   <\/span>\"\[(\lambda.f+\mu.g)'(a)=\lambda f'(a)+\mu g'(a). </li>\]\"<\/p>\n

  • la fonction \"fg\" est d\u00e9rivable en \"a\" et :\n

      <\/span>   <\/span>\"\[(fg)'(a)=f'(a)g(a)+f(a)g'(a).\]\"<\/p> <\/li>\n\n\n\n

    D\u00e9monstration<\/h3>\n\n\n\n

    <\/p>\nSoit \"(\lambda,\mu)\in\mathbb{R}^2\". Pour tout \"x\in I\setminus\{a\}\", on a :\n

      <\/span>   <\/span>\"\[\frac{(\lambda.f+\mu.g)(x)-(\lambda.f+\mu.g)(a)}{x-a} = \lambda\frac{f(x)-f(a)}{x-a}+\mu\frac{g(x)-g(a)}{x-a} \xrightarrow[x\to a]{}\lambda f'(a)+\mu g'(a).\]\"<\/p>\nDonc, \"\lambda.f+\mu.g\" est d\u00e9rivable en \"a\" et \"(\lambda.f+\mu.g)'(a)=\lambda f'(a)+\mu g'(a)\".
    \nDe plus, pour tout \"x\in I\setminus\{a\}\", on a :\n

      <\/span>   <\/span>\"\[\frac{(fg)(x)-(fg)(a)}{x-a}=f(x)\frac{g(x)-g(a)}{x-a}+g(a)\frac{f(x)-f(a)}{x-a}.\]\"<\/p>\nOr, \"f\" est d\u00e9rivable en \"a\", donc \"f\" est continue en \"a\". D’o\u00f9, par op\u00e9rations sur les limites,\n

      <\/span>   <\/span>\"\[\frac{(fg)(x)-(fg)(a)}{x-a}\xrightarrow[x\to a]{} f(a)g'(a)+g(a)f'(a).\]\"<\/p>\nDonc, \"fg\" est d\u00e9rivable en \"a\" et \"(fg)'(a)=f'(a)g(a)+f(a)g'(a).\"\n\n\n\n

    \n
    \n
    \"livre<\/figure>\n<\/div>\n\n\n\n
    <\/div>\n\n\n\n
    \n
    <\/div>\n\n\n\n

    Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

    <\/p>\n\n\n

    \n \n
    \n \n
    \n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n <\/div>\n \n
    \n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n
    \n \n\n
    <\/div>\n <\/div>\n <\/div>\n<\/div>\n \n\n
    \n 3.7\/5 - (3 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

    Vous cherchez \u00e0 calculer la d\u00e9riv\u00e9e d’une fonction ? Avec ce cours sp\u00e9cialement con\u00e7u pour cette notion, vous (…)<\/p>\n","protected":false},"author":158,"featured_media":166049,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-235481","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/235481","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=235481"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/235481\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/166049"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=235481"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=235481"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=235481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}