{"id":35,"date":"2023-09-17T11:01:09","date_gmt":"2023-09-17T11:01:09","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-de-um-produto-de-multiplicacao\/"},"modified":"2023-09-17T11:01:09","modified_gmt":"2023-09-17T11:01:09","slug":"derivada-de-um-produto-de-multiplicacao","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-de-um-produto-de-multiplicacao\/","title":{"rendered":"Derivada de um produto (ou multiplica\u00e7\u00e3o)"},"content":{"rendered":"<p>Neste artigo explicamos como derivar o produto de duas fun\u00e7\u00f5es (f\u00f3rmula). Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos de derivadas de produtos de fun\u00e7\u00f5es e at\u00e9 praticar com exerc\u00edcios resolvidos sobre derivadas de multiplica\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-de-un-producto\"><\/span> F\u00f3rmula para a derivada de um produto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A derivada de um produto de duas fun\u00e7\u00f5es diferentes \u00e9 igual ao produto da derivada da primeira fun\u00e7\u00e3o pela segunda fun\u00e7\u00e3o indiferenciada mais o produto da primeira fun\u00e7\u00e3o indiferenciada pela derivada da segunda fun\u00e7\u00e3o.<\/strong><\/p>\n<p> Em outras palavras, se <em>f(x)<\/em> e <em>g(x)<\/em> s\u00e3o duas fun\u00e7\u00f5es diferentes, a f\u00f3rmula para a derivada da multiplica\u00e7\u00e3o entre as duas fun\u00e7\u00f5es \u00e9 a seguinte: <\/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\/derive-dun-produit.webp\" alt=\"derivado de um produto\" class=\"wp-image-2103\" width=\"318\" height=\"298\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Assim, aplicando a regra da derivada de um produto, passamos de uma simples multiplica\u00e7\u00e3o para dois produtos diferentes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-la-derivada-de-un-producto\"><\/span> Exemplos de derivada de um produto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de sabermos qual \u00e9 a f\u00f3rmula da derivada de um produto (ou multiplica\u00e7\u00e3o), resolveremos v\u00e1rios exemplos desse tipo de derivada. Isso tornar\u00e1 muito mais f\u00e1cil entender como um produto de duas fun\u00e7\u00f5es \u00e9 derivado.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 1<\/h3>\n<p> Neste exemplo resolveremos a derivada de duas fun\u00e7\u00f5es potenciais multiplicando:<\/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-ac06a60a36e2b2b8b42a4e84aae6d78f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5x^2\\cdot (x^3+4x-6)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"200\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como vimos na se\u00e7\u00e3o anterior, a f\u00f3rmula para a derivada da multiplica\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-775fe6e5ac196e5a44c840866e35062d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=f(x)\\cdot g(x) \\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=f'(x)\\cdot g(x)+f(x)\\cdot g'(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"252\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, devemos primeiro calcular a derivada de cada fun\u00e7\u00e3o separadamente: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-39\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9f6ba571ede98526688967d6db6b708d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ 5x^2=10x\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"104\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-81f76a053e0ede02d46b917b5733f0cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ (x^3+4x-6)=3x^2+4\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"209\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> E uma vez que conhecemos a derivada de cada fun\u00e7\u00e3o, podemos aplicar a f\u00f3rmula da derivada do produto de duas fun\u00e7\u00f5es. Ou seja, multiplicamos a derivada do primeiro fator pelo segundo fator sem diferenciar, depois somamos o produto do primeiro fator sem diferenciar pela derivada do segundo fator:<\/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-00f424cf1f72c1d3822c14d49873253e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=5x^2\\cdot (x^3+4x-6)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] f'(x)=10x\\cdot (x^3+4x-6)+5x^2\\cdot (3x^2+4)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por fim, realizamos as opera\u00e7\u00f5es para simplificar o resultado obtido:<\/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-48b8d455b68b87932ca3a437f5ffe3a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp; =10x\\cdot (x^3+4x-6)+5x^2\\cdot (3x^2+4)\\\\[1.5ex] &amp; = 10x^4+40x^2-60x +15x^4+20x^2 \\\\[1.5ex] &amp; = 25x^4+60x^2-60x\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 2<\/h3>\n<p> Neste caso derivaremos o produto de uma constante por uma 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-0eac27878f19facde1912b0e4c80f7c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=7\\cdot (x^2+3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"151\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A regra da derivada de um produto \u00e9 a seguinte:<\/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-775fe6e5ac196e5a44c840866e35062d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=f(x)\\cdot g(x) \\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=f'(x)\\cdot g(x)+f(x)\\cdot g'(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"252\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Assim, derivamos separadamente cada fun\u00e7\u00e3o que faz parte do produto: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-42\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7e80b2c2aeed333ee755f36592771ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ 7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"67\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ce4329169c51d065b3eaa2539afa18d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ (x^2+3x)=2x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"171\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> E ent\u00e3o aplicamos a regra para a derivada de uma multiplica\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-6c2d81edaa002aeb66ca6eec22bec001_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=7\\cdot (x^2+3x)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] f'(x)=0\\cdot (x^2+3x)+7\\cdot (2x+3)=14x+21\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"353\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Observe que a derivada de uma constante \u00e9 sempre zero, portanto podemos deduzir que <strong>a derivada da multiplica\u00e7\u00e3o de uma constante por uma fun\u00e7\u00e3o \u00e9 igual ao produto da constante pela derivada da fun\u00e7\u00e3o.<\/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-df912584fe52a7417fef5fa910376453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{c}z(x)=k\\cdot f(x) \\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=k\\cdot f'(x)\\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 3<\/h3>\n<p> Vamos resolver o produto entre uma fun\u00e7\u00e3o exponencial e um logaritmo natural:<\/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-e835022b92bf8922fece3bfff5b0fe79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=4^{3x}\\cdot \\ln(x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"141\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A f\u00f3rmula para a derivada de uma multiplica\u00e7\u00e3o de duas fun\u00e7\u00f5es \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-775fe6e5ac196e5a44c840866e35062d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}z(x)=f(x)\\cdot g(x) \\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=f'(x)\\cdot g(x)+f(x)\\cdot g'(x)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"252\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Devemos, portanto, primeiro fazer separadamente a derivada de cada fun\u00e7\u00e3o que forma o produto, que s\u00e3o as seguintes: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-45\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8bd255b76b5292f44a9af4bda726a928_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ 4^{3x}=4^{3x}\\cdot \\ln (4) \\cdot 3\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"169\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5e20aacbfc682c8afa58940cf8306f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{d}{dx}\\ \\ln(x^2)=\\cfrac{2x}{x^2}=\\cfrac{2}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"156\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> O produto derivado das fun\u00e7\u00f5es \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-9e1cb417a69252fe05883f7963bcb8db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=4^{3x}\\cdot \\ln(x^2)\\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] f'(x)=4^{3x}\\cdot \\ln (4) \\cdot 3\\cdot \\ln(x^2) +4^{3x}\\cdot \\cfrac{2}{x} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"290\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivada-de-un-producto\"><\/span> Exerc\u00edcios resolvidos sobre a derivada de um produto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Deriva os seguintes produtos funcionais: <\/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-67f07c0bd975c646a55e36e71cf4a4c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) }f(x)=5\\ln(3x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" 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-d8a0c3d9bc420193e3c1990868382d3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B) }f(x)=(4x^2+1)(6x^3-7)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"225\" 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-61bc2db7b665db51e7467b0084d16de8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) }f(x)=\\text{cos}(4x)\\cdot e^{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"175\" 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-c6e9fe99ab311488946ed15068a12ea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) }f(x)=(3x^3-4x^2+8x)\\cdot \\sqrt{6x^2+3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"310\" 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-88a563e417f4c63f256a599945356dd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) }f(x)=5^{4x}\\cdot \\log_9(x^3-x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"213\" 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-aa9abf01257a07e3610d58665ff5a149_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{F) }f(x)=\\left(10x^6-6x^5\\right)^4\\cdot \\text{arcsen}(x^2+9x)\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"323\" style=\"vertical-align: -7px;\"><\/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-a2b0cfd5d2fab9c30534aaf5a8873bba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A) } f'(x)=5\\cdot \\cfrac{3}{3x} =\\cfrac{5}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"168\" 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-f46e303cda3be6c3781f7ee4c46c1680_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{B) }f'(x)&amp;=8x\\cdot (6x^3-7)+(4x^2+1)\\cdot 18x^2\\\\[1.2ex]&amp;=48x^4-56x+72x^4+18x^2\\\\[1.2ex]&amp;=120x^4+18x^2-56x \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"94\" width=\"331\" style=\"vertical-align: 0px;\"><\/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-1ee37a1f2ba1f12185bf7ea63668b7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C) }f'(x) =-4\\text{sen}(4x)\\cdot e^{x^2}+\\text{cos}(4x)\\cdot e^{x^2}\\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"351\" 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-384d51cd4aff14037124c89ca3f77ee3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D) }f'(x)=(9x^2-8x+8)\\cdot \\sqrt{6x^2+3x}+(3x^3-4x^2+8x)\\cdot\\cfrac{12x+3}{2\\sqrt{6x^2+3x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"553\" style=\"vertical-align: -16px;\"><\/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-88020b03c9d895b81bf29a7cdeda7528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E) }f'(x)=5^{4x}\\cdot \\ln(5) \\cdot 4 \\cdot \\log_9(x^3-x)+ 5^{4x}\\cdot\\cfrac{3x^2-1}{(x^3-x)\\ln(9)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"455\" 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-5dbaa6f333ff27c717b6478d26154025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\text{F) }f'(x)=&amp; 4\\left(10x^6-6x^5\\right)^3\\cdot (60x^5-30x^4)\\cdot \\text{arcsen}(x^2+9x)\\ +\\\\[1.2ex] &amp;+\\left(10x^6-6x^5\\right)^4\\cdot \\cfrac{2x+9}{\\sqrt{1-\\left(x^2+9x\\right)^2}}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"473\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-un-producto-de-tres-funciones\"><\/span> Derivado de um produto de tr\u00eas fun\u00e7\u00f5es<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, deixamos voc\u00eas com a f\u00f3rmula da derivada da multiplica\u00e7\u00e3o de 3 fun\u00e7\u00f5es, pois \u00e9 muito semelhante \u00e0 de 2 fun\u00e7\u00f5es e pode ser \u00fatil em alguns casos.<\/p>\n<p> A <strong>derivada de um produto de tr\u00eas fun\u00e7\u00f5es<\/strong> \u00e9 igual ao produto da derivada da primeira fun\u00e7\u00e3o e das outras duas fun\u00e7\u00f5es, mais o produto da derivada da segunda fun\u00e7\u00e3o e das outras duas fun\u00e7\u00f5es, mais o produto da derivada da terceira function.function pelas outras duas fun\u00e7\u00f5es.<\/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-ab98569b058580b87eb57088447f4f49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=1mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{c}z(x)=f(x)\\cdot g(x)\\cdot h(x) \\\\[1.5ex]\\color{orange}\\bm{\\downarrow}\\\\[1.5ex] z'(x)=f'(x)\\cdot g(x)\\cdot h(x)+f(x)\\cdot g'(x)\\cdot h(x)+f(x)\\cdot g(x)\\cdot h'(x)\\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por exemplo, se quisermos derivar a seguinte multiplica\u00e7\u00e3o de tr\u00eas fun\u00e7\u00f5es diferentes:<\/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-f6710d72c4b3c57f51a1855a274e4faf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x\\cdot e^{2x} \\cdot \\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"174\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para resolver a derivada devemos aplicar a regra da derivada do produto de tr\u00eas fun\u00e7\u00f5es, 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-5c925055bae8204915ed1a7ce6fea5cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3\\cdot e^{2x} \\cdot \\text{sen}(x)+3x\\cdot 2e^{2x} \\cdot \\text{sen}(x)+3x\\cdot e^{2x} \\cdot \\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"454\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"demostracion-de-la-formula-de-la-derivada-de-un-producto\"><\/span> Demonstra\u00e7\u00e3o da f\u00f3rmula da derivada de um produto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Por fim, demonstraremos a f\u00f3rmula da derivada de uma multiplica\u00e7\u00e3o. N\u00e3o \u00e9 preciso memorizar, mas \u00e9 sempre bom entender de onde v\u00eam as f\u00f3rmulas. \ud83d\ude42<\/p>\n<p> Da defini\u00e7\u00e3o matem\u00e1tica da derivada:<\/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-dc1699622d128f888c1f20599aeccf60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"219\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Seja a fun\u00e7\u00e3o <em>z<\/em> o produto de duas fun\u00e7\u00f5es diferentes:<\/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-7221f2fe5a3cdb96186c0c3bac490818_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z(x)=f(x)\\cdot g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o a derivada de <em>z<\/em> , conforme a defini\u00e7\u00e3o, ser\u00e1:<\/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-2883cdf5f77c221fda1e72fd5f69ad33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{\\bigl[f(x+h)\\cdot g(x+h)\\bigr]-\\bigl[f(x)\\cdot g(x)\\bigr]}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"371\" style=\"vertical-align: -13px;\"><\/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-0245613adf61bb881d95fdd9f88cf16f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)\\cdot g(x+h)-f(x)\\cdot g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"342\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Como sabemos, se somarmos um termo por adi\u00e7\u00e3o e subtra\u00e7\u00e3o, isso n\u00e3o afeta o resultado, desde que ambos sejam o mesmo termo. Podemos, portanto, passar para a pr\u00f3xima etapa:<\/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-2dc1543688ce758352ae824c04f44766_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)\\cdot g(x+h)\\color{orange}\\bm{-f(x+h)\\cdot g(x)+f(x+h)\\cdot g(x)}\\color{black}-f(x)\\cdot g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"696\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Agora usamos as propriedades do limite para separar o limite anterior em dois limites diferentes:<\/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-79db4573036eb08e40df29d50582ab22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)\\cdot g(x+h)-f(x+h)\\cdot g(x)}{h}+\\lim_{h \\to 0}\\frac{f(x+h)\\cdot g(x)-f(x)\\cdot g(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"582\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Extra\u00edmos o fator comum no numerador das duas fra\u00e7\u00f5es:<\/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-6e21b5ddadb3c3767bd0292a4b25a471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}\\frac{f(x+h)\\bigl(g(x+h)-g(x)\\bigr)}{h}+\\lim_{h \\to 0}\\frac{g(x)\\bigl(f(x+h)-f(x)\\bigr)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"526\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Por outro lado, conhecemos o resultado do seguinte limite:<\/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-682c9bad316f22a6e608611a63af1dbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{h \\to 0}f(x+h)=f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"155\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Podemos, portanto, simplificar os limites:<\/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-9df631d571432bd2bbe55b94bb6cb608_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=\\lim_{h \\to 0}f(x+h)\\lim_{h \\to 0}\\frac{g(x+h)-g(x)}{h}+\\lim_{h \\to 0}g(x)\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"563\" style=\"vertical-align: -13px;\"><\/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-77c3c708c8dcd5ee55fb3dfc24900eef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=f(x)\\lim_{h \\to 0}\\frac{g(x+h)-g(x)}{h}+g(x)\\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"468\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Finalmente, olhando para os dois limites restantes, cada um corresponde \u00e0 defini\u00e7\u00e3o da derivada de uma fun\u00e7\u00e3o. A igualdade pode, portanto, ser simplificada:<\/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-6d25565e3730d9f761ed2f01b1522946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle z'(x)=f(x)\\cdot g'(x)+g(x)\\cdot f'(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"252\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ou equivalente:<\/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-b1e429dfae9a3659f78a3851759c6320_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      z'(x)=f'(x)\\cdot g(x)+f(x)\\cdot g'(x) \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos como derivar o produto de duas fun\u00e7\u00f5es (f\u00f3rmula). Al\u00e9m disso, voc\u00ea poder\u00e1 ver diversos exemplos de derivadas de produtos de fun\u00e7\u00f5es e at\u00e9 praticar com exerc\u00edcios resolvidos sobre derivadas de multiplica\u00e7\u00e3o. F\u00f3rmula para a derivada de um produto A derivada de um produto de duas fun\u00e7\u00f5es diferentes \u00e9 igual ao produto da &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-de-um-produto-de-multiplicacao\/\"> <span class=\"screen-reader-text\">Derivada de um produto (ou multiplica\u00e7\u00e3o)<\/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-35","post","type-post","status-publish","format-standard","hentry","category-derivados"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Derivada de um produto (f\u00f3rmula e exerc\u00edcios resolvidos)<\/title>\n<meta name=\"description\" content=\"Explicamos como derivar o produto ou multiplica\u00e7\u00e3o de fun\u00e7\u00f5es (f\u00f3rmula). 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