{"id":380,"date":"2023-07-03T19:15:07","date_gmt":"2023-07-03T19:15:07","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/"},"modified":"2023-07-03T19:15:07","modified_gmt":"2023-07-03T19:15:07","slug":"derivada-da-funcao-exponencial","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/","title":{"rendered":"Derivada da fun\u00e7\u00e3o exponencial"},"content":{"rendered":"<p>Neste artigo explicamos como derivar uma fun\u00e7\u00e3o exponencial. Voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada exponencial (com base a e base e) e exerc\u00edcios resolvidos para derivadas de fun\u00e7\u00f5es exponenciais.<\/p>\n<p> <strong>A regra para a derivada da fun\u00e7\u00e3o exponencial depende da base da pot\u00eancia<\/strong> , pois dependendo se a base \u00e9 um n\u00famero qualquer (a) ou o n\u00famero e, a fun\u00e7\u00e3o deriva de forma diferente. \u00c9 por isso que examinaremos cada caso separadamente a seguir e, em seguida, resumiremos as duas f\u00f3rmulas para entender completamente como derivar uma fun\u00e7\u00e3o exponencial. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-la-funcion-exponencial-de-base-a\"><\/span> Derivada da fun\u00e7\u00e3o exponencial com base a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>derivada da fun\u00e7\u00e3o exponencial com base a<\/strong> \u00e9 igual ao produto da fun\u00e7\u00e3o e o logaritmo natural da base da pot\u00eancia e a derivada 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-9e08a04b92ffd23b3c62e687287db57e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=a^u\\cdot \\ln(a) \\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"393\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por exemplo, a derivada da seguinte fun\u00e7\u00e3o exponencial \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-eca1e7d1e4ab3de9246f94860ef319ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5^{x^2+1} \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=5^{x^2+1}\\cdot \\ln(5) \\cdot 2x\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"445\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-la-funcion-exponencial-de-base-e\"><\/span> Derivada da fun\u00e7\u00e3o exponencial com base e<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>derivada da fun\u00e7\u00e3o exponencial com base e<\/strong> equivale ao produto da mesma fun\u00e7\u00e3o pela derivada 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-de605bc131ee2baa1c99471c42466903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=e^u\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"340\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por exemplo, a derivada do n\u00famero e elevado a 4x \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-ea7ab035947d16b1bdb178d7626dbfbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{4x}\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=e^{4x} \\cdot 4=4e^{4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"404\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-derivada-exponencial\"><\/span> F\u00f3rmula derivada exponencial<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Como vimos, a derivada de uma fun\u00e7\u00e3o exponencial depende da sua base. E as duas f\u00f3rmulas usadas para derivar as fun\u00e7\u00f5es exponenciais s\u00e3o: <\/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-exponentielle.webp\" alt=\"derivada exponencial\" class=\"wp-image-1818\" width=\"374\" height=\"264\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-exponencial-de-e-a-la-x\"><\/span> Derivada exponencial de e a x<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos o que \u00e9 a f\u00f3rmula da derivada exponencial, analisaremos o caso da derivada de e em x, porque \u00e9 um caso curioso.<\/p>\n<p> A <strong>derivada da fun\u00e7\u00e3o e sobre x sempre resulta na pr\u00f3pria fun\u00e7\u00e3o<\/strong> , ou seja, n\u00e3o importa quantas vezes derivamos a fun\u00e7\u00e3o e <sup>x<\/sup> , sempre obteremos a mesma 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-8e988fea0397345c314c2ebc81b0ae37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} f(x)=e^x \\\\[2ex] f'(x)=e^x\\\\[2ex] f''(x)=e^x\\\\[2ex] f'''(x)=e^x\\\\ \\vdots\\\\ f^n(x)=e^x\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"184\" width=\"86\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Esta propriedade da fun\u00e7\u00e3o e elevada a x se deve ao fato da derivada de x ser 1. Portanto, ao derivar, sempre multiplicamos a pr\u00f3pria fun\u00e7\u00e3o por 1 e, como resultado, sempre obtemos a fun\u00e7\u00e3o d&#8217;origem. <\/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-813ea4b6f7f7e4fe8b42bbcfb51c7852_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^x \\quad\\longrightarrow\\quad f'(x)=e^x\\cdot 1= e^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"290\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-derivadas-de-funciones-exponenciales\"><\/span> Problemas resolvidos de derivadas de fun\u00e7\u00f5es exponenciais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Derive a seguinte fun\u00e7\u00e3o exponencial: <\/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-7de1da2b9fd94b2dc66e4b8d76e9c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o \u00e9 baseada em um n\u00famero diferente de e, ent\u00e3o precisamos usar a seguinte f\u00f3rmula:<\/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-9e08a04b92ffd23b3c62e687287db57e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=a^u\\cdot \\ln(a) \\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"393\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A derivada da fun\u00e7\u00e3o exponencial na base 3 \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-cde0dfbeae96c0d86be1974d1e9f5fa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3^x \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=3^x\\cdot \\ln(3) \\cdot 1=3^x\\cdot \\ln(3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"477\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Calcule a derivada da seguinte fun\u00e7\u00e3o exponencial: <\/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-2b67ede3af479a1a20bb869d00d7a55d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=7^{3x^2-4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" 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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o neste exerc\u00edcio \u00e9 baseada em um n\u00famero diferente de e, portanto a seguinte f\u00f3rmula deve ser aplicada:<\/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-9e08a04b92ffd23b3c62e687287db57e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=a^u\\cdot \\ln(a) \\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"393\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o a derivada da 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-db999a17327fb3983942af23c5a11a1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=7^{3x^2-4x} \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=7^{3x^2-4x}\\cdot \\ln(7) \\cdot (6x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"518\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Encontre a derivada da seguinte fun\u00e7\u00e3o exponencial com base e: <\/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-9c2e32c1eea83d0ac98d8317339d4d40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{(5x^2-9x)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"130\" 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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o neste exerc\u00edcio tem como base o n\u00famero e, portanto podemos usar a seguinte f\u00f3rmula:<\/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-de605bc131ee2baa1c99471c42466903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=e^u\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"340\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E a deriva\u00e7\u00e3o da fun\u00e7\u00e3o exponencial d\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-c0281155c7414dc3d2876048f52b3cf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=e^{(5x^2-9x)^3} \\cdot 3(5x^2-9x)^2\\cdot (10x-9)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"331\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Observe que para resolver esta derivada precisamos usar a regra da cadeia.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Encontre a derivada da seguinte fun\u00e7\u00e3o exponencial com uma raiz como 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-991dfad4c35630b1aaa73dc919c43013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=9^{\\sqrt{5x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"93\" style=\"vertical-align: -5px;\"><\/p>\n<\/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\/\">derivada de uma fun\u00e7\u00e3o radical<\/a><\/span> <\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Embora haja uma express\u00e3o radical no expoente, ainda precisamos usar a regra para derivar a fun\u00e7\u00e3o exponencial da base a:<\/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-9e08a04b92ffd23b3c62e687287db57e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=a^u\\cdot \\ln(a) \\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"393\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A derivada da fun\u00e7\u00e3o exponencial composta \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-a51d0a3b53cf4bfbf1aed0095d290002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=9^{\\sqrt{5x}}\\cdot \\ln(9) \\cdot \\cfrac{5}{2\\sqrt{5x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"206\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Derive a seguinte fun\u00e7\u00e3o exponencial da base e 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-de0492ce5da22dff9013f236e778d71d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{\\frac{x^2}{5-3x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/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-um-quociente-de-divisao\/\">derivada de um quociente de fun\u00e7\u00f5es<\/a><\/span> <\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A base da pot\u00eancia \u00e9 o n\u00famero e, ent\u00e3o usaremos a seguinte regra para dividir 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-de605bc131ee2baa1c99471c42466903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^u \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=e^u\\cdot u'\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"340\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A derivada da fun\u00e7\u00e3o exponencial \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-288902dc45d9104ecf869510b1977ee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=e^{\\frac{x^2}{5-3x}} \\cdot \\cfrac{2x\\cdot (5-3x)-x^2\\cdot (-3)}{(5-3x)^2}\\\\[3ex] &amp;=e^{\\frac{x^2}{5-3x}} \\cdot \\cfrac{10x-6x^2+3x^2}{(5-3x)^2}\\\\[3ex] &amp;=e^{\\frac{x^2}{5-3x}} \\cdot \\cfrac{10x-3x^2}{(5-3x)^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"196\" width=\"302\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos como derivar uma fun\u00e7\u00e3o exponencial. Voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada exponencial (com base a e base e) e exerc\u00edcios resolvidos para derivadas de fun\u00e7\u00f5es exponenciais. A regra para a derivada da fun\u00e7\u00e3o exponencial depende da base da pot\u00eancia , pois dependendo se a base \u00e9 um n\u00famero qualquer (a) ou o &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\"> <span class=\"screen-reader-text\">Derivada da fun\u00e7\u00e3o exponencial<\/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-380","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>Derivada da fun\u00e7\u00e3o exponencial - Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivada da fun\u00e7\u00e3o exponencial - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Neste artigo explicamos como derivar uma fun\u00e7\u00e3o exponencial. Voc\u00ea encontrar\u00e1 a f\u00f3rmula da derivada exponencial (com base a e base e) e exerc\u00edcios resolvidos para derivadas de fun\u00e7\u00f5es exponenciais. A regra para a derivada da fun\u00e7\u00e3o exponencial depende da base da pot\u00eancia , pois dependendo se a base \u00e9 um n\u00famero qualquer (a) ou o &hellip; Derivada da fun\u00e7\u00e3o exponencial Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-03T19:15:07+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e08a04b92ffd23b3c62e687287db57e_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Derivada da fun\u00e7\u00e3o exponencial\",\"datePublished\":\"2023-07-03T19:15:07+00:00\",\"dateModified\":\"2023-07-03T19:15:07+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\"},\"wordCount\":575,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Derivados\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\",\"url\":\"https:\/\/mathority.org\/pt\/derivada-da-funcao-exponencial\/\",\"name\":\"Derivada da fun\u00e7\u00e3o exponencial - 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