{"id":38,"date":"2023-09-17T10:59:38","date_gmt":"2023-09-17T10:59:38","guid":{"rendered":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/"},"modified":"2023-09-17T10:59:38","modified_gmt":"2023-09-17T10:59:38","slug":"diferenciabilidade-de-uma-funcao","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/","title":{"rendered":"Diferenciabilidade de uma fun\u00e7\u00e3o"},"content":{"rendered":"<p>Neste artigo voc\u00ea aprender\u00e1 como estudar a diferenciabilidade de uma fun\u00e7\u00e3o, ou seja, se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Al\u00e9m disso, veremos a rela\u00e7\u00e3o entre diferenciabilidade e continuidade de uma fun\u00e7\u00e3o. E, finalmente, estudaremos a diferenciabilidade de uma fun\u00e7\u00e3o por partes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-y-continuidad-de-una-funcion\"><\/span> Diferenciabilidade e continuidade de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A continuidade e a diferenciabilidade<\/strong> de uma fun\u00e7\u00e3o 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> No entanto, o inverso deste teorema \u00e9 falso: 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 a partir de sua representa\u00e7\u00e3o gr\u00e1fica:<\/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-15\">\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, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <span style=\"color:#1976d2;\"><strong>Ponto de suaviza\u00e7\u00e3o<\/strong><\/span> em x=0:<br \/> fun\u00e7\u00e3o cont\u00ednua e diferenci\u00e1vel neste est\u00e1gio. <\/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, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <span style=\"color:#1976d2;\"><strong>Ponto angular<\/strong><\/span> em x=2:<br \/> fun\u00e7\u00e3o cont\u00ednua, mas n\u00e3o diferenci\u00e1vel neste est\u00e1gio. <\/p>\n<\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"derivabilidad-de-una-funcion-a-trozos\"><\/span> Derivabilidade de uma fun\u00e7\u00e3o por partes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Assim que conhecermos a rela\u00e7\u00e3o entre continuidade e diferenciabilidade de uma fun\u00e7\u00e3o, veremos como estudar a diferenciabilidade de uma fun\u00e7\u00e3o definida por partes.<\/p>\n<p> Voc\u00ea 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 dedut\u00edvel 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 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<\/div>\n<div style=\"background-color:#FFFDE7; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> <strong>Nota:<\/strong> Para que uma fun\u00e7\u00e3o seja diferenci\u00e1vel num ponto, a fun\u00e7\u00e3o deve ser cont\u00ednua nesse ponto. Portanto, antes de calcular as derivadas laterais, precisamos de garantir que a fun\u00e7\u00e3o \u00e9 cont\u00ednua nesse ponto. Se voc\u00ea n\u00e3o sabe como se estuda a continuidade em um ponto, pode ver como se faz no seguinte link:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/continuidade-de-funcao-continua-de-uma-funcao\/\">continuidade de uma fun\u00e7\u00e3o em um ponto<\/a><\/span><\/p>\n<\/div>\n<p> Agora vamos ver um exemplo de como calcular a 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 definida 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 das duas partes 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 <strong>a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto x=2.<\/strong><\/p>\n<p> Assim que soubermos que a fun\u00e7\u00e3o \u00e9 cont\u00ednua em x=2, estudaremos a diferenciabilidade da fun\u00e7\u00e3o nesse ponto. Para fazer isso, <strong>calculamos as derivadas laterais<\/strong> da fun\u00e7\u00e3o definida em 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 n\u00e3o existiria neste ponto.<\/p>\n<p> Por fim, basta lembrar que este procedimento tamb\u00e9m \u00e9 v\u00e1lido para estudar a diferenciabilidade de uma fun\u00e7\u00e3o de valor absoluto, uma vez que fun\u00e7\u00f5es de valor absoluto tamb\u00e9m podem ser definidas por partes. Voc\u00ea pode ver como converter uma fun\u00e7\u00e3o de valor absoluto em partes aqui:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcoes-com-valor-absoluto\/\">como definir por partes uma fun\u00e7\u00e3o com valor absoluto<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-derivabilidad-de-una-funcion\"><\/span> Exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Estude a continuidade e a diferenciabilidade da seguinte fun\u00e7\u00e3o 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-3656065bb8de98bd07da153f26fd326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^3-4x^2 + 5 &amp; \\text{si} &amp;  x<1 \\\\[2ex] -x^2+3x &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"265\" style=\"vertical-align: 0px;\"><\/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-left\"> As fun\u00e7\u00f5es das duas partes s\u00e3o cont\u00ednuas, mas devemos ver se a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto cr\u00edtico x=1. 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-cf8ad0b4baa312a5ae1bb073e5c8ff8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^-} f(x) = \\lim\\limits_{x\\to 1^-} \\bigl(x^3-4x^2 + 5\\bigr)=1^3-4\\cdot 1^2 + 5=2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"413\" 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-f25bd83439fbb5cbd148b8be88f8770b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^+} f(x) = \\lim\\limits_{x\\to 1^+} \\bigl( -x^2+3x \\bigr)=-1^2+3\\cdot 1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"364\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os dois limites laterais no ponto cr\u00edtico d\u00e3o o mesmo resultado, ent\u00e3o a fun\u00e7\u00e3o \u00e9 cont\u00ednua em x=1.<\/p>\n<p class=\"has-text-align-left\"> Uma vez sabendo que a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto cr\u00edtico, estudaremos se ela \u00e9 diferenci\u00e1vel no mesmo ponto. Portanto, calculamos as derivadas laterais:<\/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-42451fa799527167fe9a2e2259248870_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 3x^2-8x  &amp; \\text{si} &amp;  x<1 \\\\[2ex] -2x+3 &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"240\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E avaliamos as duas derivadas laterais em x=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-c9afcac5ff2f762d2b471725d4e755fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-)=3\\cdot1^2-8\\cdot 1=3-8=-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"272\" 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-dd2d9e21e09ecc434cafbf5fb0b5afaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^+)=-2\\cdot 1+3=-2+3 =1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As derivadas laterais n\u00e3o coincidem no ponto x=1 ent\u00e3o a fun\u00e7\u00e3o n\u00e3o \u00e9 diferenci\u00e1vel neste 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-73b9cb3dada6f8aea03ffc1342d0f22f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-) \\neq f'(1^+) \\ \\longrightarrow \\ \\cancel{\\exists} \\ f'(1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" 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> Analise a diferenciabilidade e continuidade da seguinte fun\u00e7\u00e3o definida nas se\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-0d118e3904c810abd15e427e9c7d0504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} \\sqrt{4x} &amp; \\text{si} &amp;  x\\leq 1 \\\\[2ex] 2+\\ln x &amp; \\text{si} &amp; x> 1 \\end{array} \\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;65&#8243; width=&#8221;226&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/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-left\"> As fun\u00e7\u00f5es das duas se\u00e7\u00f5es s\u00e3o cont\u00ednuas em seus intervalos, mas tamb\u00e9m \u00e9 necess\u00e1rio saber se a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto cr\u00edtico de mudan\u00e7a de defini\u00e7\u00e3o x=1. Portanto, definimos os limites laterais da fun\u00e7\u00e3o neste 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-2c4fb7ef0ee9b3feeb5e15654528fc71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^-} f(x) = \\lim\\limits_{x\\to 1^-} \\sqrt{4x} = \\sqrt{4\\cdot 1} = \\sqrt{4}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"323\" 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-e509a70bd0e356f44ccac7ba6f075f8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 1^+} f(x) = \\lim\\limits_{x\\to 1^+} \\bigl( 2+\\ln x \\bigr) = 2 + \\ln (1) = 2+0 =2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"399\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os dois limites laterais no ponto cr\u00edtico d\u00e3o o mesmo resultado, ent\u00e3o a fun\u00e7\u00e3o \u00e9 cont\u00ednua em x=1.<\/p>\n<p class=\"has-text-align-left\"> E agora estudamos se a fun\u00e7\u00e3o \u00e9 deriv\u00e1vel neste ponto calculando as derivadas laterais:<\/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-8261f3d268b47d9171710997c8cc70bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} \\cfrac{4}{2\\sqrt{4x}}  &amp; \\text{si} &amp;  x<1 \\\\[4ex] \\cfrac{1}{x} &amp; \\text{si} &amp; x\\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"217\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Avaliamos as duas derivadas laterais em x=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-de4ee71a175b34be07061d47470afe0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-)=\\cfrac{4}{2\\sqrt{4\\cdot1}}=\\cfrac{4}{2\\sqrt{4}}=\\cfrac{4}{2\\cdot 2}=\\cfrac{4}{4}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"309\" 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-594a8d05da432ab8962ff799d62d25a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^+)=\\cfrac{1}{1}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"115\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As derivadas laterais s\u00e3o iguais, ent\u00e3o a fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel em x=1 e o valor da derivada \u00e9 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-a58ba22b03193839f5070da4e2c1faf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1^-) = f'(1^+) = 1 \\ \\longrightarrow \\ \\bm{f'(1) = 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"274\" 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> Determine se a seguinte fun\u00e7\u00e3o por partes \u00e9 cont\u00ednua e diferenci\u00e1vel em todo o seu dom\u00ednio:<\/p>\n<pre class=\"ql-errors\"> *** QuickLaTeX cannot compile formula:\n\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^2+2x+1 &amp; \\text{si} &amp; x\\leq -1 \\\\[2ex] 2x+2 &amp; \\text{ si} &amp; -1&lt;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\"&gt;&lt;div class=\"otfm-sp__title\"&gt; &lt;strong&gt;View solution&lt;\/strong&gt;&lt;\/div&gt;&lt; \/div&gt; The functions of all three parts are continuous, but we still need to check if the function is continuous at critical points. We therefore first check the continuity of the function at the point x=-1 by solving the lateral limits at this point:\n\n*** Error message:\nMissing $ inserted.\nleading text: \\displaystyle\nMissing { inserted.\nleading text: ...=\"wp-block-otfm-box-spoiler-start otfm-sp__\nMissing { inserted.\nleading text: ...ox-spoiler-start otfm-sp__wrapper otfm-sp__\nMissing { inserted.\nleading text: ...m-sp__wrapper otfm-sp__box js-otfm-sp-box__\nMissing { inserted.\nleading text: ...fm-sp__box js-otfm-sp-box__closed otfm-sp__\nYou can't use `macro parameter character #' in math mode.\nleading text: ...=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#\nMissing { inserted.\nleading text: ...e=\"text-align:center\"&gt;&lt;div class=\"otfm-sp__\nPlease use \\mathaccent for accents in math mode.\nleading text: ...g&gt;&lt;\/div&gt;&lt;\/div&gt; The functions of the three parts\nPlease use \\mathaccent for accents in math mode.\nleading text: ...are continuous, but we still need to see\n\n<\/pre>\n<p> \\lim\\limits_{x\\to -1^-} f(x) = \\lim\\limits_{x\\to -1^-} \\bigl(x^2+2x+1\\bigr) = (-1)^ 2+2(-1)+1 =0 \\lim\\limits_{x\\to -1^+} f(x) = \\lim\\limits_{x\\to -1^+} \\bigl(2x+2\\bigr ) = 2(-1)+2=0<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fadd0ce26a497a6a6c73bfaa7ed28f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" Les deux limites lat\u00e9rales au point x=-1 donnent le m\u00eame r\u00e9sultat, donc la fonction est continue en x=-1. Nous allons maintenant v\u00e9rifier si la fonction est continue ou non au point x=2 : \" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"765\" style=\"vertical-align: 0px;\"><\/p>\n<p> \\lim\\limits_{x\\to 2^-} f(x) = \\lim\\limits_{x\\to 2^-} \\bigl(2x+2\\bigr) = 2\\cdot 2+2=4+2= 6 \\lim\\limits_{x\\to 2^+} f(x) = \\lim\\limits_{x\\to 2^+} \\bigl( -x^2+8x\\bigr) = -2^2+8\\ cponto 2 = -4+16=12<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c93f9d5e367031de798abaf833523710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" En revanche, les limites lat\u00e9rales au point x=2 ne donnent pas le m\u00eame r\u00e9sultat, donc la fonction n'est pas continue en x=2. De plus, comme il n'est pas continu \u00e0 ce stade, il ne sera pas non plus d\u00e9rivable \u00e0 x=2. Une fois que l'on a \u00e9tudi\u00e9 la continuit\u00e9 de la fonction, on passe \u00e0 la diff\u00e9rentiabilit\u00e9. On calcule donc les d\u00e9riv\u00e9es lat\u00e9rales :\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"908\" style=\"vertical-align: -4px;\"><\/p>\n<p> \\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 2x+2 &amp; \\text{si} &amp; x\\leq -1 \\\\[2ex] 2 &amp; \\text{si} &amp; -1<\/p>\n<p class=\"has-text-align-left\"> J\u00e1 sabemos que a fun\u00e7\u00e3o n\u00e3o \u00e9 diferenci\u00e1vel em x=2, ent\u00e3o s\u00f3 precisamos estudar se a fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel em x=-1. Para fazer isso, avaliamos as duas derivadas laterais 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-cafb8bbe438865e050973664b6915fa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^-)=2(-1)+2 = -2+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"272\" 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-5781d213b36777be5b2611a84ca95e96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^+)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"95\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As derivadas laterais n\u00e3o coincidem no ponto x=-1, ent\u00e3o a fun\u00e7\u00e3o n\u00e3o \u00e9 diferenci\u00e1vel nesse 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-f3674d70c292d967d7074a0b4bee230e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1^-) \\neq f'(-1^+) \\ \\longrightarrow \\ \\cancel{\\exists} \\ f'(-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"267\" 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 4<\/h3>\n<p> Calcule o valor dos par\u00e2metros a e b para que a seguinte fun\u00e7\u00e3o por partes seja cont\u00ednua e diferenci\u00e1vel em todo o seu dom\u00ednio: <\/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-ce34d5d8a949fb3a0b904e9bf7d32f5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} 2e^{x-3} + a &amp; \\text{si} &amp;  x< 3 \\\\[2ex](x-b)^2 &amp; \\text{si} &amp; x\\geq 3 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"243\" style=\"vertical-align: 0px;\"><\/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-left\"> Quaisquer que sejam os valores das inc\u00f3gnitas, a fun\u00e7\u00e3o \u00e9 cont\u00ednua e diferenci\u00e1vel em todos os pontos exceto em x=3, onde sua continuidade e diferenciabilidade devem ser verificadas.<\/p>\n<p class=\"has-text-align-left\"> Para que a fun\u00e7\u00e3o seja cont\u00ednua num ponto, os dois limites laterais nesse ponto devem coincidir. Portanto, avaliamos os limites laterais 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-21920afd0fe6c6a35983a39034b3f9f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 3^-} f(x) = \\lim\\limits_{x\\to 3^-} \\bigl(2e^{x-3}+a\\bigr) = 2e^{3-3}+a = 2 \\cdot e^0+a =2\\cdot 1 +a = 2+a\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"566\" 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-f527d4602a1bdc26d763df1064b956d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x\\to 3^+} f(x) = \\lim\\limits_{x\\to 3^+} (x-b)^2 = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"282\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os dois valores obtidos nos limites laterais devem, portanto, ser iguais para que a fun\u00e7\u00e3o seja cont\u00ednua:<\/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-3bec8303f39289b7f2cd7e6e439703c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Analisaremos agora a diferenciabilidade no ponto x=3. Encontramos as derivadas laterais:<\/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-d542fc9488644f0c144059ae1403d961_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f'(x)= \\left\\{ \\begin{array}{lcl} 2e^{x-3}  &amp; \\text{si} &amp;  x< 3 \\\\[2ex]2(x-b) &amp; \\text{si} &amp; x\\geq 3 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"236\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E avaliamos as duas derivadas laterais 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-0fb3afef2352f0f5f96302b987a5de9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3^-)= 2e^{3-3} =  2e^0 = 2\\cdot 1 = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"250\" 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-360474b477f347569ad1e3b64b63cc79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3^+)=2(3-b) = 6 - 2b\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, para que a fun\u00e7\u00e3o seja diferenci\u00e1vel em x=3, os valores obtidos das derivadas laterais devem ser iguais:<\/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-8a98a45230845ad86846cd7db486af0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=6-2b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvendo esta equa\u00e7\u00e3o podemos encontrar o valor de b: <\/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-7f53d30cae8270029d25ea28322b6986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2b=6-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"79\" 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-c263311ac7433ce2b417bb7ad0ef449b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2b=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" 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-7dd2f0fd5e5207a815bf5789ada67541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=\\cfrac{4}{2} =\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, uma vez conhecido o valor do par\u00e2metro b, podemos calcular o valor do par\u00e2metro a resolvendo a equa\u00e7\u00e3o que obtivemos anteriormente nos limites laterais: <\/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-3bec8303f39289b7f2cd7e6e439703c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6468721c373f078ad3e97a290c2d86f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a = (3-2)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" 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-d1cc01602cd205cae7b9c9b8ba391760_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2+a =1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"72\" style=\"vertical-align: -2px;\"><\/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-ce22cc1bba3ee9612a7f8cb2624d2483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a =1-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"72\" 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-be5f9e4b074e8b3ba5d0e96f8ae4e2cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a =-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" 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 voc\u00ea aprender\u00e1 como estudar a diferenciabilidade de uma fun\u00e7\u00e3o, ou seja, se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Al\u00e9m disso, veremos a rela\u00e7\u00e3o entre diferenciabilidade e continuidade de uma fun\u00e7\u00e3o. E, finalmente, estudaremos a diferenciabilidade de uma fun\u00e7\u00e3o por partes. Diferenciabilidade e continuidade de uma fun\u00e7\u00e3o A continuidade e a diferenciabilidade de uma &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\"> <span class=\"screen-reader-text\">Diferenciabilidade de uma fun\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-38","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 Diferenciabilidade de uma fun\u00e7\u00e3o (teoria e exerc\u00edcios resolvidos)<\/title>\n<meta name=\"description\" content=\"Como estudar a diferenciabilidade de uma fun\u00e7\u00e3o e saber se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Com exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o por partes.\" \/>\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\/diferenciabilidade-de-uma-funcao\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Diferenciabilidade de uma fun\u00e7\u00e3o (teoria e exerc\u00edcios resolvidos)\" \/>\n<meta property=\"og:description\" content=\"Como estudar a diferenciabilidade de uma fun\u00e7\u00e3o e saber se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Com exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o por partes.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T10:59:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-pour-representer-une-fonction-quadratique-incomplete.webp\" \/>\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=\"8 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Diferenciabilidade de uma fun\u00e7\u00e3o\",\"datePublished\":\"2023-09-17T10:59:38+00:00\",\"dateModified\":\"2023-09-17T10:59:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\"},\"wordCount\":1278,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Derivados\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\",\"url\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\",\"name\":\"\u25b7 Diferenciabilidade de uma fun\u00e7\u00e3o (teoria e exerc\u00edcios resolvidos)\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-09-17T10:59:38+00:00\",\"dateModified\":\"2023-09-17T10:59:38+00:00\",\"description\":\"Como estudar a diferenciabilidade de uma fun\u00e7\u00e3o e saber se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Com exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o por partes.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Diferenciabilidade de uma fun\u00e7\u00e3o\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u25b7 Diferenciabilidade de uma fun\u00e7\u00e3o (teoria e exerc\u00edcios resolvidos)","description":"Como estudar a diferenciabilidade de uma fun\u00e7\u00e3o e saber se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Com exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o por partes.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/","og_locale":"pt_BR","og_type":"article","og_title":"\u25b7 Diferenciabilidade de uma fun\u00e7\u00e3o (teoria e exerc\u00edcios resolvidos)","og_description":"Como estudar a diferenciabilidade de uma fun\u00e7\u00e3o e saber se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Com exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o por partes.","og_url":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/","article_published_time":"2023-09-17T10:59:38+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-pour-representer-une-fonction-quadratique-incomplete.webp"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"8 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Diferenciabilidade de uma fun\u00e7\u00e3o","datePublished":"2023-09-17T10:59:38+00:00","dateModified":"2023-09-17T10:59:38+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/"},"wordCount":1278,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Derivados"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/","url":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/","name":"\u25b7 Diferenciabilidade de uma fun\u00e7\u00e3o (teoria e exerc\u00edcios resolvidos)","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-09-17T10:59:38+00:00","dateModified":"2023-09-17T10:59:38+00:00","description":"Como estudar a diferenciabilidade de uma fun\u00e7\u00e3o e saber se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel ou n\u00e3o. Com exerc\u00edcios resolvidos sobre a diferenciabilidade de uma fun\u00e7\u00e3o por partes.","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/diferenciabilidade-de-uma-funcao\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Diferenciabilidade de uma fun\u00e7\u00e3o"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=38"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/38\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=38"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}