{"id":401,"date":"2023-07-03T00:18:18","date_gmt":"2023-07-03T00:18:18","guid":{"rendered":"https:\/\/mathority.org\/pt\/derivados\/"},"modified":"2023-07-03T00:18:18","modified_gmt":"2023-07-03T00:18:18","slug":"derivados","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/derivados\/","title":{"rendered":"Derivados"},"content":{"rendered":"<p>Aqui explicamos como derivar todos os tipos de fun\u00e7\u00f5es. Voc\u00ea encontrar\u00e1 as f\u00f3rmulas para todas as derivadas acompanhadas de exemplos e exerc\u00edcios passo a passo de derivadas. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formules-derivees.webp\" alt=\"f\u00f3rmulas derivadas\" class=\"wp-image-2945\" width=\"226\" height=\"226\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-las-derivadas\"><\/span> O que s\u00e3o produtos derivados?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Derivadas<\/strong> s\u00e3o regras matem\u00e1ticas usadas para estudar fun\u00e7\u00f5es. Em particular, a <strong>derivada de uma fun\u00e7\u00e3o num ponto<\/strong> \u00e9 o resultado de um limite e indica o comportamento da fun\u00e7\u00e3o nesse ponto.<\/p>\n<p> A derivada de uma fun\u00e7\u00e3o \u00e9 expressa com o sinal primo <em>&#8216;<\/em> , ou seja, a fun\u00e7\u00e3o <em>f'(x)<\/em> \u00e9 a derivada da fun\u00e7\u00e3o <em>f(x)<\/em> .<\/p>\n<p> Geometricamente, o significado da derivada de uma fun\u00e7\u00e3o num ponto \u00e9 a inclina\u00e7\u00e3o da tangente \u00e0 fun\u00e7\u00e3o nesse ponto. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equation-de-la-tangente-ligne.webp\" alt=\"significado de derivados\" class=\"wp-image-2306\" width=\"392\" height=\"391\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> A <strong>defini\u00e7\u00e3o matem\u00e1tica da derivada de uma fun\u00e7\u00e3o<\/strong> \u00e9 a seguinte:<\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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> No entanto, a derivada de uma fun\u00e7\u00e3o geralmente n\u00e3o \u00e9 calculada usando a f\u00f3rmula acima, mas aplicam-se regras de diferencia\u00e7\u00e3o dependendo do tipo de fun\u00e7\u00e3o. Todas as f\u00f3rmulas de deriva\u00e7\u00e3o s\u00e3o explicadas abaixo.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formulas-de-las-derivadas\"><\/span>f\u00f3rmulas derivadas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de ver a defini\u00e7\u00e3o de derivadas, veremos como elas s\u00e3o feitas, explicando cada tipo de derivada com um exemplo. O objetivo deste post \u00e9 que voc\u00ea entenda bem o conceito de derivadas, ent\u00e3o se no final voc\u00ea tiver alguma d\u00favida sobre como uma fun\u00e7\u00e3o \u00e9 derivada, pode nos perguntar nos coment\u00e1rios.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-una-constante\"><\/span>derivado de uma constante<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>derivada de uma constante<\/strong> \u00e9 sempre zero, independentemente do valor da constante.<\/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-c7bd8f1aee171f251c313218820e22f0_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=1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      f(x)=k \\quad \\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=0 \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, para encontrar a derivada de uma fun\u00e7\u00e3o constante, n\u00e3o h\u00e1 necessidade de fazer nenhuma matem\u00e1tica, apenas a derivada \u00e9 zero.<\/p>\n<p> D\u00ea uma olhada nos seguintes exemplos pr\u00e1ticos de derivadas de constantes: <\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-561a1b2c2b0347c0cb38ed7565e46fa7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=3 \\qquad \\longrightarrow\\qquad f'(x)=0\\\\[3ex]g(x)=-5 \\qquad \\longrightarrow\\qquad g'(x)=0\\\\[3ex]h(x)=291 \\qquad \\longrightarrow\\qquad h'(x)=0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"265\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-una-funcion-lineal\"><\/span> Derivada de uma fun\u00e7\u00e3o linear<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>derivada de uma fun\u00e7\u00e3o linear<\/strong> \u00e9 o coeficiente do termo de primeiro grau, ou seja, a derivada de uma fun\u00e7\u00e3o linear <em>f(x)=Ax+B<\/em> \u00e9 igual a <em>A<\/em><\/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-c55a9a25283e37ab61dc79856ee92a11_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=1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      f(x)=Ax+B\\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=A \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> D\u00ea uma olhada nos seguintes exemplos de como esse tipo de fun\u00e7\u00e3o foi derivado: <\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-53d0d8c8814ef6884b442c3c50cce8a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}f(x)=3x-1\\quad\\longrightarrow\\quad f'(x)=3\\\\[3ex]f(x)=5x\\quad\\longrightarrow\\quad f'(x)=5\\\\[3ex] f(x)=-2x+9\\quad\\longrightarrow\\quad f'(x)=-2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"280\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-una-potencia\"><\/span> derivado de um poder<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>derivada de uma pot\u00eancia<\/strong> , ou fun\u00e7\u00e3o potencial, \u00e9 o produto do expoente da pot\u00eancia vezes a base elevada ao expoente menos 1.<\/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-1e7df9c631129b040e262f67f36b41be_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=1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      f(x)=x^k \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=k\\cdot x^{k-1} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, para derivar uma pot\u00eancia, basta multiplicar a fun\u00e7\u00e3o pelo expoente e subtrair uma unidade do expoente.<\/p>\n<p> Por exemplo, a derivada da pot\u00eancia x ao cubo \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-0c0dd56d2e4a99c896f5e035d51f80be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3 \\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=3\\cdot x^{3-1}=3x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"404\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Voc\u00ea pode praticar exerc\u00edcios (e mais dif\u00edceis) desse tipo de derivada aqui:<\/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-potencial-de-potencia\/\">exerc\u00edcios resolvidos para a derivada de uma pot\u00eancia<\/a><\/span><\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-una-raiz\"><\/span> derivado de uma raiz<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><meta charset=\"utf-8\"> A <strong>derivada de uma raiz,<\/strong> ou fun\u00e7\u00e3o irracional, \u00e9 igual a um dividido pelo produto do \u00edndice da raiz vezes a mesma raiz subtraindo 1 do expoente do radicando.<\/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-8b8e85735674a043d4fb2c448038ceb8_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=1pt, boxsep=1.5mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      f(x)=\\sqrt[n]{x}\\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=\\cfrac{1}{n\\sqrt[n]{x^{n-1}}} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como exemplo, abaixo voc\u00ea pode ver a derivada da raiz quadrada de x resolvida:<\/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-5e879d493e8b67755617d2aed1743cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\sqrt{x}\\quad\\color{orange}\\bm{\\longrightarrow}\\color{black}\\quad f'(x)=\\cfrac{1}{2\\sqrt{x^{2-1}}}=\\cfrac{1}{2\\sqrt{x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"425\" style=\"vertical-align: -16px;\"><\/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\/\">exerc\u00edcios resolvidos para a derivada de uma raiz<\/a><\/span> <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-una-funcion-exponencial\"><\/span> Derivada de uma fun\u00e7\u00e3o exponencial<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>derivada de uma fun\u00e7\u00e3o exponencial<\/strong> depende se a base \u00e9 o n\u00famero <em>e<\/em> ou outro n\u00famero. Existem portanto duas f\u00f3rmulas para derivar este tipo de fun\u00e7\u00e3o e deve-se utilizar aquela que corresponde de acordo com a base de 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-c21e61c79c41a4f27d53a41495521bdd_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=1pt, boxsep=1.8mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{l}f(x)=a^x \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=a^x\\cdot \\ln(a)\\\\[3ex] f(x)=e^x \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=e^x \\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Abaixo voc\u00ea pode ver duas derivadas resolvidas deste tipo de fun\u00e7\u00f5es: <\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-90bd19942de37daaf7af04179eaf5e91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=7^{x} \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=7^x\\cdot \\ln(7)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"364\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-71dd62c13ea22caa62fb0e8af338fbdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^{x} \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=e^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"312\" 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-da-funcao-exponencial\/\">exerc\u00edcios resolvidos para a derivada de uma fun\u00e7\u00e3o exponencial<\/a><\/span> <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivada-de-una-funcion-logaritmica\"><\/span> Derivada de uma fun\u00e7\u00e3o logar\u00edtmica<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>derivada de uma fun\u00e7\u00e3o logar\u00edtmica<\/strong> depende da base do logaritmo, pois se o logaritmo for natural deve-se aplicar uma f\u00f3rmula para encontrar a derivada e se o logaritmo tiver outro n\u00famero como base deve-se utilizar outra regra.<\/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-d384075ba6ad6dcfaf82949d33ad397b_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=1pt, boxsep=1.8mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{l}f(x)=\\ln(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x}\\\\[3ex] f(x)=\\log_a(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x\\cdot\\ln(a)}\\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por exemplo, a derivada do logaritmo de base tr\u00eas de x \u00e9:<\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-e61ac2c2f66f8b05dce8760bdec17d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_3(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{x\\cdot\\ln(3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"397\" style=\"vertical-align: -17px;\"><\/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-logaritmica-logaritmo-natural-neperiano\/\">exerc\u00edcios resolvidos para a derivada de uma fun\u00e7\u00e3o logar\u00edtmica<\/a><\/span><\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivadas-trigonometricas\"><\/span>Derivadas trigonom\u00e9tricas<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> As tr\u00eas principais <strong>derivadas trigonom\u00e9tricas<\/strong> s\u00e3o a derivada da fun\u00e7\u00e3o seno, a fun\u00e7\u00e3o cosseno e a fun\u00e7\u00e3o tangente, cujas f\u00f3rmulas s\u00e3o as seguintes:<\/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-a9e9d0157c2b1eb994571ba96aae4f26_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=1pt, boxsep=1.8mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{l}f(x)=\\text{sen}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cos}(x)\\\\[2.5ex] f(x)=\\text{cos}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=-\\text{sen}(x)\\\\[1.1ex]f(x)=\\text{tan}(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\cfrac{1}{\\text{cos}^2(x)}\\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Logicamente, existem v\u00e1rios tipos de fun\u00e7\u00f5es trigonom\u00e9tricas, como secante, cossecante, cotangente, fun\u00e7\u00f5es trigonom\u00e9tricas hiperb\u00f3licas, fun\u00e7\u00f5es trigonom\u00e9tricas inversas, etc. Mas as regras mais utilizadas para drifting s\u00e3o as tr\u00eas acima.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"reglas-de-derivacion\"><\/span> regras de refer\u00eancia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Quando temos opera\u00e7\u00f5es com fun\u00e7\u00f5es, as derivadas s\u00e3o resolvidas de forma diferente. Para isso, precisamos utilizar as <strong>regras de diferencia\u00e7\u00e3o<\/strong> , que nos permitem derivar adi\u00e7\u00e3o, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o e divis\u00e3o de 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-ba4f3225344df68c84b4437ecb0c7536_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=1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\begin{array}{l}z(x)=f(x)\\pm g(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} z'(x)=f'(x)\\pm g'(x)\\\\[4ex] z(x)=f(x)\\cdot g(x) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} z'(x)=f'(x)\\cdot g(x)+f(x)\\cdot g'(x)\\\\[4ex]z(x)=\\cfrac{f(x)}{g(x)} \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} z'(x)=\\cfrac{f'(x)\\cdot g(x)-f(x)\\cdot g'(x)}{\\bigl(g(x)\\bigr)^2}\\end{array} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, para resolver derivadas com opera\u00e7\u00f5es, n\u00e3o precisamos apenas aplicar as regras das derivadas, mas tamb\u00e9m usar a f\u00f3rmula para cada tipo de derivada.<\/p>\n<p> Para que voc\u00ea veja como encontrar esse tipo de derivada, resolveremos v\u00e1rios exerc\u00edcios a seguir:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\"><u style=\"text-decoration-color:#FF9B28;\">Derivada de uma soma:<\/u><\/span> <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-efca27bad0818b86e6e42ee15a31ed6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x^2+5x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-8a8dcfca37df757df3fd79292ead67b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=6x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>Como voc\u00ea pode ver, para resolver a derivada de toda a fun\u00e7\u00e3o, a f\u00f3rmula da derivada de uma pot\u00eancia foi aplicada a cada termo da soma.<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\"><u style=\"text-decoration-color:#FF9B28;\">Derivado de um produto:<\/u><\/span> <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-8c539c8db8dbf532a639e09af47a583a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=4^{x}\\cdot \\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A derivada do primeiro termo do produto \u00e9 4 <sup>x<\/sup> ln(4), e a derivada do seno \u00e9 o cosseno. Portanto, a derivada da multiplica\u00e7\u00e3o \u00e9: <\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-5829940bb4219334d7e238b40a19794e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=4^{x}\\cdot \\ln (4) \\cdot \\text{sen}(x) +4^{x}\\cdot \\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"290\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\"><u style=\"text-decoration-color:#FF9B28;\">Derivada de um quociente:<\/u><\/span> <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-59b390fee61ab3c2cbb4dc2230386658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^3+4x^2}{5x^2-8}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"127\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> No numerador e denominador da fra\u00e7\u00e3o temos um polin\u00f4mio, ent\u00e3o para obter a derivada precisamos usar a f\u00f3rmula da derivada de um quociente, a f\u00f3rmula da derivada de uma adi\u00e7\u00e3o (ou subtra\u00e7\u00e3o) e a f\u00f3rmula da derivada de tem poder: <\/p>\n<p class=\"has-text-align-center\"><meta charset=\"utf-8\"><\/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-af3f7cb513883d1fa5dadca23701c19d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}f'(x)&amp;=\\cfrac{(3x^2+8x)\\cdot (5x^2-8)-(x^3+4x^2)\\cdot 10x}{\\left(5x^2-8\\right)^2}\\\\[2ex]&amp;=\\cfrac{15x^4-24x^2+40x^3-64x-10x^4-40x^3}{25x^4+64-80x^2}\\\\[2ex]&amp;=\\cfrac{5x^4-24x^2-64x}{25x^4-80x^2+64}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"178\" width=\"379\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-de-la-cadena\"><\/span> Regra da cadeia<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><meta charset=\"utf-8\"> A <strong>regra da cadeia<\/strong> \u00e9 uma f\u00f3rmula usada para derivar fun\u00e7\u00f5es compostas. A regra da cadeia afirma que a derivada de uma fun\u00e7\u00e3o composta <em>f(g(x))<\/em> \u00e9 igual \u00e0 derivada <em>f'(g(x))<\/em> multiplicada pela derivada <em>g'(x)<\/em> .<\/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-662d8c44c904e83267bbca5f968ca546_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=1pt, boxsep=1.5mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      z(x)=f\\bigl(g(x)\\bigr) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black}z'(x)=f'\\bigl(g(x)\\bigr)\\cdot g'(x) \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Essa no\u00e7\u00e3o de derivadas geralmente \u00e9 mais dif\u00edcil de assimilar, por isso resolveremos passo a passo um exerc\u00edcio como exemplo:<\/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-c1863b1f92befa398b5c8692d239abf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sen}(x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Efetivamente, \u00e9 uma composi\u00e7\u00e3o de fun\u00e7\u00f5es porque temos a fun\u00e7\u00e3o x <sup>3<\/sup> dentro da fun\u00e7\u00e3o seno, portanto, devemos utilizar a regra da cadeia para encontrar a derivada da fun\u00e7\u00e3o composta.<\/p>\n<p> Por um lado, a derivada do seno \u00e9 o cosseno, ent\u00e3o a derivada da fun\u00e7\u00e3o exterior ser\u00e1 o cosseno com o mesmo argumento do seno:<\/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-2046d85a1440fe95dccc4d8bb553e2f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\\bigl(g(x)\\bigr)=\\text{sen}(x^3) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'\\bigl(g(x)\\bigr)=\\text{cos}(x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"441\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> E, por outro lado, calculamos a derivada de x <sup>3<\/sup> usando a f\u00f3rmula da 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-1d045f948b3519322ae6771bd4497d70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)=x^3\\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} g'(x)=3x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"320\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Assim, a derivada da fun\u00e7\u00e3o composta inteira \u00e9 o produto das duas derivadas:<\/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-ab755de02fa9196320c59676d77cd2e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{sen}(x^3) \\quad\\color{orange}\\bm{\\longrightarrow}\\quad\\color{black} f'(x)=\\text{cos}(x^3)\\cdot 3x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"430\" 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\/cadeia-de-regras-derivada\/\">exerc\u00edcios de derivadas resolvidos com a regra da cadeia<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-de-una-funcion\"><\/span> Diferenciabilidade de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A <strong>continuidade e a diferenciabilidade de uma fun\u00e7\u00e3o<\/strong> em um ponto est\u00e3o relacionadas da seguinte forma:<\/p>\n<ul>\n<li> Se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel num ponto, a fun\u00e7\u00e3o \u00e9 cont\u00ednua nesse ponto.<\/li>\n<li> Se uma fun\u00e7\u00e3o n\u00e3o \u00e9 cont\u00ednua num ponto, tamb\u00e9m n\u00e3o \u00e9 diferenci\u00e1vel nesse ponto.<\/li>\n<\/ul>\n<p> Contudo, o inverso deste teorema \u00e9 falso, ou seja, s\u00f3 porque uma fun\u00e7\u00e3o \u00e9 cont\u00ednua num ponto n\u00e3o significa que seja sempre diferenci\u00e1vel nesse ponto.<\/p>\n<p> Voc\u00ea tamb\u00e9m pode ver se uma fun\u00e7\u00e3o \u00e9 ou n\u00e3o diferenci\u00e1vel em um ponto de seu gr\u00e1fico:<\/p>\n<ul>\n<li> Se for um <strong>ponto suave,<\/strong> a fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel neste ponto.<\/li>\n<li> Se for um <strong>ponto angular,<\/strong> a fun\u00e7\u00e3o \u00e9 cont\u00ednua, mas n\u00e3o diferenci\u00e1vel neste ponto. <\/li>\n<\/ul>\n<div class=\"wp-block-columns is-layout-flex wp-container-3\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\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\/exercices-resolus-pour-representer-une-fonction-quadratique-incomplete.webp\" alt=\"\" class=\"wp-image-140\" width=\"250\" height=\"279\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <span style=\"color:#1976d2;\"><strong>Ponto suave<\/strong><\/span> em x = 0:<br \/> fun\u00e7\u00e3o cont\u00ednua e diferenci\u00e1vel neste ponto. <\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\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\/comment-representer-graphiquement-une-fonction-avec-valeur-absolue.webp\" alt=\"\" class=\"wp-image-230\" width=\"286\" height=\"305\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <span style=\"color:#1976d2;\"><strong>Ponto inclinado<\/strong><\/span> em x=2:<br \/> fun\u00e7\u00e3o cont\u00ednua, mas n\u00e3o diferenci\u00e1vel neste ponto.<\/p>\n<\/div>\n<\/div>\n<p> Voc\u00ea tamb\u00e9m pode saber se uma fun\u00e7\u00e3o por partes \u00e9 diferenci\u00e1vel em um ponto calculando as <strong>derivadas laterais<\/strong> nesse ponto: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 20px; border-radius:20px;\">\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">Se as derivadas laterais num ponto n\u00e3o forem iguais, a fun\u00e7\u00e3o n\u00e3o \u00e9 diferenci\u00e1vel nesse ponto:<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97c60c64dc01a7e0a9084313d15b0886_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x_o^-) \\neq f'(x_o^+) \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p> N\u00e3o \u00e9 diferenci\u00e1vel em<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ab403a6241009b92035b251f86c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_o\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li> <span style=\"color:#000000;font-weight: normal;\">Se as derivadas laterais num ponto coincidem, a fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel nesse ponto:<\/span><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3f9477828318b9e6392465762f831642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x_o^-) = f'(x_o^+) \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<p> Sim, \u00e9 deriv\u00e1vel em<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67ab403a6241009b92035b251f86c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_o\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<\/div>\n<p> Agora vamos ver um exemplo de c\u00e1lculo da derivada de uma fun\u00e7\u00e3o definida por partes em um ponto:<\/p>\n<ul>\n<li> Estude a continuidade e a diferenciabilidade da seguinte fun\u00e7\u00e3o por partes no ponto x=2:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a98eee72521c68fd394eb6209a7d0a59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=  \\left\\{ \\begin{array}{lcl} 3x^2-6x &amp; \\text{si} &amp;  x<2 \\\\[2ex] 6\\ln (x-1) &amp; \\text{si} &amp; x\\geq 2 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"249\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> As fun\u00e7\u00f5es de ambas as se\u00e7\u00f5es s\u00e3o cont\u00ednuas em seus respectivos intervalos, por\u00e9m \u00e9 necess\u00e1rio verificar se a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto cr\u00edtico x=2. Para fazer isso, resolvemos os limites laterais da fun\u00e7\u00e3o no ponto:<\/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-dad5bcb0055431aa87a67068c04d2ce2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 2^-} f(x) = \\lim\\limits_{x\\to 2^-} \\bigl(3x^2-6x\\bigr) = 3\\cdot2^2-6\\cdot2=12-12=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"449\" style=\"vertical-align: -14px;\"><\/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-31e1ba2ea2c5fd9fa86e5cefed0e5535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 2^+} f(x) = \\lim\\limits_{x\\to 2^+} 6\\ln (x-1) = 6\\ln (2-1)=6 \\ln 1=6 \\cdot 0= \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"474\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Os limites laterais no ponto cr\u00edtico nos deram o mesmo resultado, ent\u00e3o a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto x=2.<\/p>\n<p> Assim que soubermos que a fun\u00e7\u00e3o \u00e9 cont\u00ednua em x=2, estudaremos a diferenciabilidade da fun\u00e7\u00e3o neste ponto. Para fazer isso, calculamos as <strong>derivadas laterais<\/strong> da fun\u00e7\u00e3o definida por partes:<\/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-3709995609d0f69f382ff651e397c00a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 6x-6 &amp; \\text{si} &amp;  x<2 \\\\[2ex] \\cfrac{6}{x-1} &amp; \\text{si} &amp; x\\geq 2 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"222\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agora avaliamos cada derivada lateral no ponto cr\u00edtico:<\/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-ca27189960575c1151402d040bfa76f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^-)=6\\cdot2-6=12-6 = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"239\" 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-058310a16d7d545ea56e99517845842b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^+)=\\cfrac{6}{2-1} = \\cfrac{6}{1} = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"181\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> As duas derivadas laterais nos deram o mesmo resultado, ent\u00e3o a fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel em x=2 e o valor da derivada \u00e9 6:<\/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-aadf046f27916c46f0a302d6e0c34113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2^-) = f'(2^+) = 6 \\ \\longrightarrow \\ \\bm{f'(2) = 6}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"275\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por outro lado, se as derivadas laterais nos tivessem dado um resultado diferente, isso significaria que a fun\u00e7\u00e3o n\u00e3o \u00e9 diferenci\u00e1vel em x=2. Em outras palavras, a derivada neste ponto n\u00e3o existiria.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\">exerc\u00edcios resolvidos para diferenciabilidade de uma fun\u00e7\u00e3o<\/a><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aqui explicamos como derivar todos os tipos de fun\u00e7\u00f5es. Voc\u00ea encontrar\u00e1 as f\u00f3rmulas para todas as derivadas acompanhadas de exemplos e exerc\u00edcios passo a passo de derivadas. O que s\u00e3o produtos derivados? Derivadas s\u00e3o regras matem\u00e1ticas usadas para estudar fun\u00e7\u00f5es. Em particular, a derivada de uma fun\u00e7\u00e3o num ponto \u00e9 o resultado de um limite &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/derivados\/\"> <span class=\"screen-reader-text\">Derivados<\/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-401","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>Derivados - 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\/derivados\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivados - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Aqui explicamos como derivar todos os tipos de fun\u00e7\u00f5es. 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