{"id":28,"date":"2023-09-17T11:03:43","date_gmt":"2023-09-17T11:03:43","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-de-uma-funcao-potencial-de-potencia\/"},"modified":"2023-09-17T11:03:43","modified_gmt":"2023-09-17T11:03:43","slug":"derivada-de-uma-funcao-potencial-de-potencia","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-de-uma-funcao-potencial-de-potencia\/","title":{"rendered":"Derivada de uma pot\u00eancia (ou fun\u00e7\u00e3o potencial)"},"content":{"rendered":"<p>Aqui explicamos como derivar uma pot\u00eancia (ou fun\u00e7\u00e3o potencial), voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada de uma pot\u00eancia, v\u00e1rios exemplos e poder\u00e1 at\u00e9 praticar com exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-una-potencia\"><\/span> F\u00f3rmula para a derivada de uma pot\u00eancia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada de uma pot\u00eancia, ou fun\u00e7\u00e3o potencial, \u00e9 o produto do expoente da pot\u00eancia vezes a base elevada ao expoente menos 1 vezes a derivada da base.<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54482f54c568a1e8f0e5706dd9ea1f58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=u^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot u^{k-1}\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, se a base for a <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcao-de-identidade\/\">fun\u00e7\u00e3o identidade<\/a><\/span> , para obter a pot\u00eancia, basta multiplicar a fun\u00e7\u00e3o pelo expoente e subtrair uma unidade do expoente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46eb5b8a4c6968ca5e37af417fd2fb31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot x^{k-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"355\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Na verdade, a derivada da fun\u00e7\u00e3o identidade \u00e9 igual a 1.<\/p>\n<p> Em resumo, para derivar uma fun\u00e7\u00e3o potencial, existem duas f\u00f3rmulas: a primeira que sempre pode ser usada e a segunda que s\u00f3 pode ser aplicada quando a base for um x. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/derivee-dune-puissance.webp\" alt=\"derivado de um poder\" class=\"wp-image-1760\" width=\"354\" height=\"264\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Podemos facilmente verificar que a primeira f\u00f3rmula apresentada para a derivada de uma pot\u00eancia \u00e9 semelhante \u00e0 segunda, mas aplicando a regra da cadeia.<\/p>\n<p> Observe que essas f\u00f3rmulas s\u00f3 podem ser usadas quando a vari\u00e1vel \u00e9 a base da pot\u00eancia, se o x estiver no denominador voc\u00ea deve aplicar a regra para a derivada de uma fun\u00e7\u00e3o exponencial:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\">derivada de uma fun\u00e7\u00e3o exponencial<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-derivadas-de-potencias\"><\/span> Exemplos de derivados de pot\u00eancia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos a f\u00f3rmula da derivada de uma fun\u00e7\u00e3o potencial, explicaremos v\u00e1rios exemplos deste tipo de derivada para que voc\u00ea entenda como as pot\u00eancias s\u00e3o derivadas. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-derivada-de-una-potencia-de-base-x\"><\/span> Exemplo 1: Derivada de uma pot\u00eancia base x<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1a77d84e8f9a9d165a989ae8edd67a7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como explicamos na se\u00e7\u00e3o anterior, quando a base da pot\u00eancia \u00e9 apenas x, a f\u00f3rmula que devemos usar para derivar a fun\u00e7\u00e3o \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46eb5b8a4c6968ca5e37af417fd2fb31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot x^{k-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"355\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A derivada da pot\u00eancia x elevada \u00e0 pot\u00eancia de 4 \u00e9, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4f99a41ebf24ffb9dc9d375925fbb03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^4 \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=4\\cdot x^{4-1}=4x^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"404\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-derivada-de-una-potencia-con-parentesis\"><\/span> Exemplo 2: Derivada de uma pot\u00eancia com par\u00eanteses<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7e7bcf1cc01f1609b200eb0707450f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=(2x-1)^5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste exemplo a base n\u00e3o \u00e9 a fun\u00e7\u00e3o identidade, devemos portanto usar a f\u00f3rmula geral para a derivada de uma pot\u00eancia:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54482f54c568a1e8f0e5706dd9ea1f58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=u^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot u^{k-1}\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A fun\u00e7\u00e3o entre par\u00eanteses \u00e9 uma fun\u00e7\u00e3o linear, ent\u00e3o sua derivada \u00e9 2. Portanto, a derivada de toda a fun\u00e7\u00e3o potencial \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b3fb5d76dac22c3fad5ebe3a0cae630_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=(2x-1)^5 \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=5\\cdot (2x-1)^4\\cdot 2=10(2x-1)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"577\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3-derivada-de-una-potencia-negativa\"><\/span> Exemplo 3: Derivada de uma pot\u00eancia negativa<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0bb49a61d47e399ec84ffddce840b4a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=(\\log 9x)^{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso temos uma fun\u00e7\u00e3o potencial cujo expoente \u00e9 negativo e cuja base \u00e9 um logaritmo, portanto utilizaremos a seguinte f\u00f3rmula para diferenciar a fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54482f54c568a1e8f0e5706dd9ea1f58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=u^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot u^{k-1}\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Mesmo que o expoente da pot\u00eancia seja negativo, tamb\u00e9m deve ser subtra\u00eddo dele. A derivada da fun\u00e7\u00e3o \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b883e28fe86b43bae53225fbfc0989c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=-2\\cdot (\\log 9x)^{-3}\\cdot \\cfrac{1}{9x\\cdot \\ln 10}\\cdot 9 =\\cfrac{-2(\\log 9x)^{-3}}{x\\ln 10}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"412\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Se tiver alguma d\u00favida sobre a solu\u00e7\u00e3o, pode consultar aqui a f\u00f3rmula da derivada de uma fun\u00e7\u00e3o logar\u00edtmica:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-funcao-logaritmica-logaritmo-natural-neperiano\/\">derivada de uma fun\u00e7\u00e3o logar\u00edtmica<\/a><\/span> <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-4-derivada-de-una-potencia-con-raiz\"><\/span> Exemplo 4: Derivada de uma pot\u00eancia com raiz<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74c457eaa374d11591afb05c08c79a2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\sqrt[3]{(5x+2)^7}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"149\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> A fun\u00e7\u00e3o neste exemplo \u00e9 uma pot\u00eancia dentro de uma express\u00e3o regular. No entanto, os radicais podem ser transformados em express\u00f5es potenciais, de modo que a fun\u00e7\u00e3o pode ser simplificada convertendo-a em uma fun\u00e7\u00e3o potencial com um expoente fracion\u00e1rio:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a851a10c275664fd40272c3f18ebb696_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=(5x+2)^{\\frac{7}{3}}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Aplicamos agora a f\u00f3rmula para a derivada de uma pot\u00eancia de uma vari\u00e1vel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54482f54c568a1e8f0e5706dd9ea1f58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=u^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot u^{k-1}\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E a derivada \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48032a3f39b346214870d2930c0e070f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{7}{3}\\cdot (5x+2)^{\\frac{7}{3}-1} \\cdot 5=\\cfrac{35}{3}(5x+2)^{\\frac{4}{3}}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"314\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Voc\u00ea tamb\u00e9m pode diferenciar esses tipos de fun\u00e7\u00f5es usando a regra da derivada raiz:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-funcao-raiz-irracional-radical\/\">derivado de uma raiz<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-de-una-potencia\"><\/span> Exerc\u00edcios resolvidos sobre a derivada de uma pot\u00eancia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Calcule a derivada das seguintes pot\u00eancias: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8439a4f480db3b28403219b3d6c617c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f(x)=x^8\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-586b1cce6da03d26fc251a34f15b8713_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f(x)=3x^5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2048513ae93bb32ac7333db56114f8f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f(x)=-2x^{-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61b9db052aa145257ef92ac543621b69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f(x)=(3x^2-4x)^6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"173\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-234c4c2a85b56e062d186e33461534c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f(x)=6(x^2+10)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-888598e68e596f15e1e4b585c2287c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f(x)=\\cfrac{1}{(9x+2)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"156\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bc2353357a484e175ac77e99e4446cb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{G) } f(x)=\\sqrt{4x^5}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"127\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e21d4a7733a95b407d4a53114146489_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{H) } \\displaystyle f(x)=\\left(x^2-4x+\\frac{1}{3}\\right)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"210\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c04e44f3e248cec297f4476885e22a25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{I) } f(x)=\\text{sen}^3(6-2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"172\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong> <\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32e3eaeb51430921053c75da627cacd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=8x^7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44e8de4cfb9cb267005e390e06c393e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) } f'(x)=15x^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4394259c55b711c8668f4d72e3bd4766_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) } f'(x)=8x^{-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a916b5238f094595e63d04a86303a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) } f'(x)=6(3x^2-4x)^5\\cdot (3x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-698c94329d2e6edc002bf2b99d4de74e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) } f'(x)=3\\cdot 6(x^2+10)^2 \\cdot 2x=36x(x^2+10)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"358\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2c94d0e23facd72c3a85241bfc4552a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) } f'(x)=-3\\cdot (9x+2)^{-4}\\cdot 9 =-27(9x+2)^{-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"363\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f488aced09f0e560280f5ce55a1b002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{G) } f'(x)=\\cfrac{5}{2}\\cdot 2x^{\\frac{5}{2}-1} =5x^{\\frac{3}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"211\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-19d1d31c8415cb654940148ec1a25835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{H) } \\displaystyle f'(x)=4\\left(x^2-4x+\\cfrac{1}{3}\\right)^3\\cdot (2x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"303\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa9c91854971313e88fde0d535c5a6c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{I) } f'(x)=3\\text{sen}^2(6-2x)\\cdot \\text{cos}(6-2x)\\cdot (-2)=-6x\\cdot \\text{sen}^2(6-2x)\\cdot \\text{cos}(6-2x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"599\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui explicamos como derivar uma pot\u00eancia (ou fun\u00e7\u00e3o potencial), voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada de uma pot\u00eancia, v\u00e1rios exemplos e poder\u00e1 at\u00e9 praticar com exerc\u00edcios resolvidos passo a passo. F\u00f3rmula para a derivada de uma pot\u00eancia A derivada de uma pot\u00eancia, ou fun\u00e7\u00e3o potencial, \u00e9 o produto do expoente da pot\u00eancia vezes a base &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-de-uma-funcao-potencial-de-potencia\/\"> <span class=\"screen-reader-text\">Derivada de uma pot\u00eancia (ou fun\u00e7\u00e3o potencial)<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","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":"","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":"","footnotes":""},"categories":[11],"tags":[],"class_list":["post-28","post","type-post","status-publish","format-standard","hentry","category-derivados"],"yoast_head":"<!-- This site is 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