{"id":29711,"date":"2024-11-27T12:00:45","date_gmt":"2024-11-27T12:00:45","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29711"},"modified":"2024-11-27T17:35:54","modified_gmt":"2024-11-27T17:35:54","slug":"a-derivada-como-o-limite-de-uma-funcao","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/a-derivada-como-o-limite-de-uma-funcao\/","title":{"rendered":"A Derivada como o Limite de uma Fun\u00e7\u00e3o"},"content":{"rendered":"<style>\np {\n  text-align: justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">A Derivada como o Limite de uma Fun\u00e7\u00e3o<\/h1>\n<p style=\"text-align:center;\">\n  <em><strong>Resumo:<\/strong> Nesta aula, exploraremos o conceito de derivada como a ferramenta matem\u00e1tica para analisar mudan\u00e7as em fun\u00e7\u00f5es. Partiremos da inclina\u00e7\u00e3o de uma reta secante e, ao calcular o limite conforme os pontos se aproximam, definiremos a derivada como a inclina\u00e7\u00e3o da reta tangente. Al\u00e9m disso, estudaremos suas propriedades principais e regras, como as de soma, produto e quociente, fundamentais para aplicar derivadas na an\u00e1lise de fun\u00e7\u00f5es e fen\u00f4menos de mudan\u00e7a.<\/em>\n<\/p>\n<p style=\"text-align:center;\"><strong>Objetivos de Aprendizagem<\/strong><\/p>\n<p>Ao final desta aula, os alunos ser\u00e3o capazes de:<\/p>\n<ol>\n<li><strong>Compreender<\/strong> a derivada como o limite que descreve a mudan\u00e7a instant\u00e2nea em uma fun\u00e7\u00e3o e como a inclina\u00e7\u00e3o da reta tangente a uma curva em um ponto.<\/li>\n<li><strong>Explicar<\/strong> como a diferenciabilidade implica continuidade em fun\u00e7\u00f5es.<\/li>\n<li><strong>Demonstrar<\/strong> as regras b\u00e1sicas de diferencia\u00e7\u00e3o a partir da defini\u00e7\u00e3o formal.<\/li>\n<li><strong>Aplicar<\/strong> as propriedades da \u00e1lgebra das derivadas (soma, produto e quociente) em problemas matem\u00e1ticos.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><\/p>\n<p><a href=\"#1\"><strong>O conceito de derivada<\/strong><\/a><br \/>\n<a href=\"#1\">A inclina\u00e7\u00e3o da reta secante<\/a><br \/>\n<a href=\"#1\">C\u00e1lculo do limite: A derivada e a inclina\u00e7\u00e3o da reta tangente<\/a><br \/>\n<a href=\"#1\">Defini\u00e7\u00e3o alternativa<\/a><br \/>\n<a href=\"#1\"><strong>Propriedades das Derivadas<\/strong><\/a><br \/>\n<a href=\"#1\">Diferenciabilidade implica continuidade<\/a><br \/>\n<a href=\"#1\">\u00c1lgebra das derivadas<\/a><\/p>\n<p><center><br \/>\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/TFxATgmYvkY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>O Conceito de Derivada<\/h2>\n<p>A natureza \u00e9, em geral, sujeita a mudan\u00e7as, e a ferramenta matem\u00e1tica por excel\u00eancia para calcular e compreender essas mudan\u00e7as \u00e9 a derivada. Ela surge ao perguntar: \u00abO que acontece com o valor de uma fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> quando a vari\u00e1vel <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u00e9 aumentada ou diminu\u00edda por uma quantidade arbitrariamente pequena <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span>?\u00bb O conceito de derivada emerge como o limite de uma fun\u00e7\u00e3o ao analisar essa quest\u00e3o.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>A Inclina\u00e7\u00e3o da Reta Secante<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=164s\" target=\"_blank\" rel=\"noopener\"><strong>Considere uma fun\u00e7\u00e3o<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> avaliada em dois pontos <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. Qualquer reta que corta dois pontos de uma curva \u00e9 chamada de \u00abreta secante\u00bb e se parece com o que aparece na figura abaixo.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"Gr\u00e1fico da reta secante\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/--KZ1YA55iug\/YI_jLiez_RI\/AAAAAAAAFCs\/xYcWyzwUaf88McAiTNK7l6tOSZQKyZFdwCLcBGAsYHQ\/s0\/graficosecante.PNG\" alt=\"Gr\u00e1fico da reta secante\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><\/noscript><\/p>\n<p>A inclina\u00e7\u00e3o desta reta secante \u00e9 dada por:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p><a name=\"3\"><\/a><\/p>\n<h3>C\u00e1lculo do Limite: A Derivada e a Inclina\u00e7\u00e3o da Reta Tangente<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=278s\" target=\"_blank\" rel=\"noopener\"><strong>Considere a reta secante da curva<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span> que passa pelos pontos <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. Ao calcular o limite quando <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> tende a zero, obtemos a reta tangente \u00e0 curva que passa por <span class=\"katex-eq\" data-katex-display=\"false\">(x_0, f(x_0)).<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"Gr\u00e1fico da reta tangente\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-8wCxY7adTBw\/YI_kfLeezzI\/AAAAAAAAFC0\/o6nKbRKv1SISYU3Rx7ML5Rly29edqey3ACLcBGAsYHQ\/s0\/grafico%2Brecta%2Btangente.PNG\" alt=\"Gr\u00e1fico da reta tangente\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><\/noscript><\/p>\n<p>A partir disso, a defini\u00e7\u00e3o formal da derivada de uma fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> em um ponto <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> \u00e9 dada como:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{df(x_0)}{dx}:= \\lim_{\\Delta x \\to 0}\\dfrac{\\Delta f(x_0)}{\\Delta x} = \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>Essa defini\u00e7\u00e3o representa a inclina\u00e7\u00e3o da reta tangente que passa por <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Defini\u00e7\u00e3o Alternativa<\/h3>\n<p>Uma forma alternativa de apresentar a defini\u00e7\u00e3o de derivada como limite \u00e9 obtida a partir da seguinte substitui\u00e7\u00e3o:<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nx_i &amp;= x_0\\\\\n\nx_f &amp;= x_i + \\Delta x\n\n\\end{array}\n\n<\/span>\n<p>Assim, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_f - x_i<\/span>, e a defini\u00e7\u00e3o de derivada fica:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\dfrac{df(x_i)}{dx} &amp;=\\displaystyle \\lim_{\\Delta x \\to 0}\\dfrac{ f(x_i + \\Delta x) - f(x_i)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{x_f - x_i \\to 0} \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\\\\ \\\\\n\n&amp;=\\displaystyle  \\lim_{x_f \\to x_i } \\dfrac{f(x_f) - f(x_i)}{x_f - x_i}\n\n\\end{array}\n\n<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"Defini\u00e7\u00e3o de derivada como o limite das inclina\u00e7\u00f5es das retas secantes\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GLyWOue8OUs\/YJAHOc_lTOI\/AAAAAAAAFC8\/3IV-onfsq9QC4nyweccS4ZN_O-JlWVz8wCLcBGAsYHQ\/s0\/definicion%2Bderivada%2Bcomo%2Blimite.PNG\" alt=\"Defini\u00e7\u00e3o de derivada como o limite das inclina\u00e7\u00f5es das retas secantes\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><\/noscript><\/p>\n<p>Ambas as defini\u00e7\u00f5es s\u00e3o equivalentes e podem ser usadas de forma intercambi\u00e1vel, dependendo da conveni\u00eancia.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Propriedades das Derivadas<\/h2>\n<p>Dizemos que uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel em <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> quando existe o seguinte limite:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x_0 + \\Delta x) - f(x_0)}{\\Delta x}<\/span>\n<p>Al\u00e9m disso, dizemos que \u00e9 diferenci\u00e1vel em um conjunto <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> se o limite est\u00e1 bem definido para todos os <span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I.<\/span> Fun\u00e7\u00f5es diferenci\u00e1veis possuem as seguintes propriedades:<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Diferenciabilidade Implica Continuidade<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=526s\" target=\"_blank\" rel=\"noopener\"><strong>Se uma fun\u00e7\u00e3o \u00e9 diferenci\u00e1vel em <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> ent\u00e3o \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span> Isso pode ser demonstrado atrav\u00e9s do seguinte argumento:<\/p>\n<p>Para que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> seja cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, \u00e9 necess\u00e1rio que:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = f(x_0)<\/span>\n<p>Analisando o lado esquerdo desta express\u00e3o, temos:<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to x_0} f(x) &amp;= \\displaystyle \\lim_{x\\to x_0} \\left[ f(x) + f(x_0) - f(x_0) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( f(x)  - f(x_0) \\right) \\right] \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{x\\to x_0} \\left[f(x_0) + \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0)  \\right] \\\\ \\\\\n\n&amp;=f(x_0) +\\displaystyle \\lim_{x\\to x_0} \\left[ \\left( \\dfrac{f(x)  - f(x_0)}{x- x_0} \\right)(x-x_0) \\right]\n\\end{array}\n\n<\/span>\n<p>Portanto, para que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> seja cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, \u00e9 necess\u00e1rio que o limite do lado direito esteja bem definido. Isso ocorre se, e somente se:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\dfrac{f(x) - f(x_0)}{x-x_0} = \\dfrac{df(x_0)}{dx}<\/span>\n<p>Em outras palavras, se <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u00e9 diferenci\u00e1vel em <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>. Consequentemente, se <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u00e9 diferenci\u00e1vel em <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, ent\u00e3o \u00e9 cont\u00ednua nesse ponto.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>\u00c1lgebra das Derivadas<\/h3>\n<p>Sejam <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> fun\u00e7\u00f5es diferenci\u00e1veis para todo <span class=\"katex-eq\" data-katex-display=\"false\">x \\in I<\/span>, e sejam <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta \\in \\mathbb{R}.<\/span> Ent\u00e3o, temos:<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha \\dfrac{df(x)}{dx} \\pm \\beta\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( f(x) g(x) \\right) = \\dfrac{df(x)}{dx} g(x) + f(x)\\dfrac{dg(x)}{dx}<\/span><\/li>\n<li>Se <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0<\/span>, ent\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left( \\dfrac{f(x)}{g(x)} \\right) = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx} }{\\left[g(x)\\right]^2}<\/span><\/li>\n<\/ol>\n<p>Como podemos observar, a \u00e1lgebra das derivadas n\u00e3o \u00e9 t\u00e3o intuitiva quanto pode parecer \u00e0 primeira vista. Entretanto, essas propriedades podem ser demonstradas sem muita dificuldade a partir da defini\u00e7\u00e3o de derivada como limite.<\/p>\n<p><span style=\"color: #000080;\">DEMONSTRA\u00c7\u00c3O:<\/span><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=925s\" target=\"_blank\" rel=\"noopener\"><strong>A demonstra\u00e7\u00e3o da derivada da soma<\/strong><\/a> segue o seguinte racioc\u00ednio:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\alpha f(x) \\pm \\beta g(x) \\right) &amp; =\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\left[\\alpha f(x+\\Delta x) \\pm \\beta g(x+ \\Delta x)\\right] - \\left[\\alpha f(x) \\pm \\beta g(x) \\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\left[\\alpha f(x+\\Delta x) - \\alpha f(x)\\right] \\pm \\left[\\beta g(x+\\Delta x) - \\beta g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\alpha \\left[ f(x+\\Delta x) -  f(x)\\right] \\pm  \\beta  \\left[ g(x+\\Delta x) - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\alpha \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) -  f(x)}{\\Delta x} \\pm \\beta \\lim_{\\Delta x \\to 0} \\dfrac{ g(x+\\Delta x) -  g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\alpha \\dfrac{df(x)}{dx} \\pm \\beta \\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=1059s\" target=\"_blank\" rel=\"noopener\"><strong>Por outro lado, a demonstra\u00e7\u00e3o da derivada do produto<\/strong><\/a> \u00e9 um pouco mais elaborada:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left[f(x)g(x)\\right] &amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) -  f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) g(x+\\Delta x) + \\color{red}f(x)g(x+\\Delta x) - f(x)g(x+\\Delta x) \\color{black} - f(x) g(x)}{\\Delta x} \\\\ \\\\\n\n&amp;= \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\left[f(x+\\Delta x) - f(x) \\right] g(x+\\Delta x) + f(x) \\left[g(x+\\Delta x)  - g(x)\\right]}{\\Delta x} \\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x) \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;=\\displaystyle \\lim_{\\Delta x \\to 0} g(x+\\Delta x)\\lim_{\\Delta x \\to 0} \\dfrac{f(x+\\Delta x) - f(x)}{\\Delta x} + f(x)\\lim_{\\Delta x \\to 0} \\dfrac{g(x+\\Delta x) - g(x)}{\\Delta x}\\\\ \\\\\n\n&amp;= g(x) \\dfrac{df(x)}{dx} + f(x)\\dfrac{dg(x)}{dx}\n\n\\end{array}\n\n<\/span>\n<p>Isso se baseia no fato de que, como <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u00e9 diferenci\u00e1vel, ela \u00e9 cont\u00ednua. Assim, <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{\\Delta x\\to 0 } g(x+\\Delta x) = g(x).<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=162s\" target=\"_blank\" rel=\"noopener\"><strong>Finalmente, para a demonstra\u00e7\u00e3o da derivada do quociente,<\/strong><\/a> podemos aproveitar o resultado da derivada do produto. Considere uma fun\u00e7\u00e3o da forma <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = f(x)\/g(x),<\/span> com <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0.<\/span> A partir disso, temos:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\dfrac{df(x)}{dx}= \\dfrac{d}{dx}(k(x)g(x)) = \\dfrac{dk(x)}{dx}g(x) + k(x)\\dfrac{dg(x)}{dx}<\/span>\n<p>Resolvendo para <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}<\/span>, obtemos:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}g(x) = \\dfrac{df(x)}{dx} - k(x)\\dfrac{dg(x)}{dx} = \\dfrac{d}{dx}f(x) - \\dfrac{f(x)}{g(x)}\\dfrac{dg(x)}{dx} <\/span>\n<p>Portanto:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)\n\n &amp;= \\dfrac{dk(x)}{dx} =\\dfrac{1}{g(x)} \\dfrac{df(x)}{dx} - \\dfrac{f(x)}{\\left[g(x)\\right]^2}\\dfrac{dg(x)}{dx} \\\\ \\\\\n\n&amp; = \\dfrac{\\dfrac{df(x)}{dx}g(x) - f(x) \\dfrac{dg(x)}{dx}}{[g(x)]^2}\n\n\\end{array}\n\n<\/span>\n<p>Isso conclui a demonstra\u00e7\u00e3o que busc\u00e1vamos.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Derivada como o Limite de uma Fun\u00e7\u00e3o Resumo: Nesta aula, exploraremos o conceito de derivada como a ferramenta matem\u00e1tica para analisar mudan\u00e7as em fun\u00e7\u00f5es. Partiremos da inclina\u00e7\u00e3o de uma reta secante e, ao calcular o limite conforme os pontos se aproximam, definiremos a derivada como a inclina\u00e7\u00e3o da reta tangente. Al\u00e9m disso, estudaremos suas [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29706,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":29,"footnotes":""},"categories":[856,571],"tags":[],"citadela-post-location":[],"class_list":["post-29711","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculo-diferencial-pt","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>A Derivada como o Limite de uma Fun\u00e7\u00e3o - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explico o conceito de derivada como um limite de uma fun\u00e7\u00e3o e ensino de forma clara e simples como deduzir todas as suas propriedades.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/pt\/a-derivada-como-o-limite-de-uma-funcao\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A Derivada como o Limite de uma Fun\u00e7\u00e3o\" \/>\n<meta property=\"og:description\" content=\"Explico o conceito de derivada como um limite de uma fun\u00e7\u00e3o e ensino de forma clara e simples como deduzir todas as suas propriedades.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/a-derivada-como-o-limite-de-uma-funcao\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-11-27T12:00:45+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-11-27T17:35:54+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/derivada014-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"A Derivada como o Limite de uma Fun\u00e7\u00e3o\" \/>\n<meta name=\"twitter:description\" content=\"Explico o conceito de derivada como um limite de uma fun\u00e7\u00e3o e ensino de forma clara e simples como deduzir todas as suas propriedades.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/derivada014.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"A Derivada como o Limite de uma Fun\u00e7\u00e3o\",\"datePublished\":\"2024-11-27T12:00:45+00:00\",\"dateModified\":\"2024-11-27T17:35:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/\"},\"wordCount\":1691,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/11\\\/derivada014.jpg\",\"articleSection\":[\"C\u00e1lculo Diferencial\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/a-derivada-como-o-limite-de-uma-funcao\\\/\",\"name\":\"A Derivada como o Limite de uma Fun\u00e7\u00e3o - 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