{"id":33359,"date":"2024-11-30T13:00:38","date_gmt":"2024-11-30T13:00:38","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33359"},"modified":"2025-07-19T10:36:53","modified_gmt":"2025-07-19T10:36:53","slug":"derivadas-de-polinomios-funcoes-trigonometricas-e-logaritmos","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/derivadas-de-polinomios-funcoes-trigonometricas-e-logaritmos\/","title":{"rendered":"Derivadas de polin\u00f4mios, fun\u00e7\u00f5es trigonom\u00e9tricas e logaritmos"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<style><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><br \/>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Derivadas de polin\u00f4mios, trigonom\u00e9tricas e logaritmo<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\nA derivada \u00e9 uma ferramenta central do c\u00e1lculo diferencial, com aplica\u00e7\u00f5es fundamentais nas ci\u00eancias, na engenharia e na economia. Este artigo oferece um guia progressivo para dominar a diferencia\u00e7\u00e3o de fun\u00e7\u00f5es, desde polin\u00f4mios at\u00e9 fun\u00e7\u00f5es trigonom\u00e9tricas e logar\u00edtmicas. Por meio de demonstra\u00e7\u00f5es e exemplos concretos, busca-se compreender tanto a aplica\u00e7\u00e3o das regras quanto seu fundamento.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objetivos de Aprendizagem<\/strong><\/p>\n<ol>\n<li><strong>Compreender<\/strong> o conceito geral de derivada e suas propriedades fundamentais.<\/li>\n<li><strong>Aplicar<\/strong> a defini\u00e7\u00e3o formal de derivada para calcular derivadas b\u00e1sicas.<\/li>\n<li><strong>Demonstrar<\/strong> por meio de limites a derivada de fun\u00e7\u00f5es constantes e da fun\u00e7\u00e3o identidade.<\/li>\n<li><strong>Obter<\/strong> as regras para derivar fun\u00e7\u00f5es trigonom\u00e9tricas a partir das derivadas fundamentais do seno e do cosseno.<\/li>\n<li><strong>Calcular<\/strong> derivadas de fun\u00e7\u00f5es trigonom\u00e9tricas compostas utilizando regras alg\u00e9bricas.<\/li>\n<li><strong>Demonstrar<\/strong> formalmente a derivada do logaritmo natural por meio de limites.<\/li>\n<\/ol>\n<p style=\"text-align:center\">\n<strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">Derivada das fun\u00e7\u00f5es alg\u00e9bricas<\/a><br \/>\n<a href=\"#2\">Derivadas das fun\u00e7\u00f5es transcendentes<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" title=\"YouTube video player\" data-src=\"https:\/\/www.youtube.com\/embed\/LudVXLRnmLw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>At\u00e9 agora revisamos apenas o que \u00e9 a derivada e algumas de suas propriedades alg\u00e9bricas, mas nada foi dito sobre como calcul\u00e1-las. Aqui resolveremos esse problema mostrando cada uma das t\u00e9cnicas de diferencia\u00e7\u00e3o e como elas s\u00e3o obtidas em cada caso.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Derivada das fun\u00e7\u00f5es alg\u00e9bricas<\/h2>\n<h3>Fun\u00e7\u00e3o constante<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=104s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = c,<\/span> sendo <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> <\/strong><\/a>uma constante real qualquer, ent\u00e3o teremos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{d}{dx}c = 0<\/span>\n<p><span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span> Na verdade, essa demonstra\u00e7\u00e3o \u00e9 feita em um \u00fanico passo:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}c &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{c - c}{\\Delta x} \\quad \\text{; Def. da derivada de $f(x)=c$} \\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{0}{\\Delta x} = 0\n\n\\end{array}\n\n<\/span>\n<h3>Fun\u00e7\u00e3o identidade<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x,<\/span> ent\u00e3o:<\/span><\/strong><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx}{dx}=1<\/span>\n<p><span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span> Quase igual \u00e0 anterior, tamb\u00e9m se resolve em um \u00fanico passo:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x &amp;= \\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{(x+\\Delta x) - x}{\\Delta x} \\quad \\text{; Def. da derivada para $f(x) = x$}\\\\ \\\\\n\n&amp; &amp;=\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\Delta x}{\\Delta x} = 1\n\n\\end{array}\n\n<\/span>\n<h3>Pot\u00eancias naturais<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=239s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x^n,<\/span> onde<\/span> <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u00e9 um n\u00famero natural qualquer, ent\u00e3o teremos que:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} =\\frac{dx^n}{dx} =nx^{n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span> Para demonstrar este teorema, teremos que usar o <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" target=\"_blank\" rel=\"noopener\">teorema do bin\u00f4mio de Newton<\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp;\\displaystyle \\dfrac{d}{dx}x^n = \\lim_{\\Delta x \\to 0} \\frac{(x+\\Delta x)^n -x^n}{\\Delta x} &amp;\\text{ ; Def. de limite para $f(x)= x^n$} \\\\ \\\\\n\n&amp; \\displaystyle \\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=0}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{; Teorema do Bin\u00f4mio de Newton, sobre $(1)$} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle x^n + \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right] - x^n}{\\Delta x} &amp; \\text{; Separando o primeiro termo da soma} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\dfrac{\\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k} \\right]}{\\Delta x} &amp; \\text{; Cancelando termos semelhantes} \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[\\sum_{k=1}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\\\ \\\\\n\n&amp; \\displaystyle\\phantom{\\displaystyle \\dfrac{d}{dx}x^n} =\\lim_{\\Delta x \\to 0} \\displaystyle \\left[ {{n}\\choose{1}} x^{n-1}(\\Delta x)^{0} + \\sum_{k=2}^n {{n}\\choose{k}} x^{n-k}(\\Delta x)^{k-1} \\right] &amp; \\text{; Extraindo o primeiro termo da soma} \\\\ \\\\\n\n&amp; \\displaystyle \\color{blue} {\\displaystyle \\dfrac{d}{dx}x^n} = n x^{n-1} &amp; \\color{black}\n\n\\end{array}\n\n<\/span>\n<h3>Pot\u00eancias Inteiras<\/h3>\n<p>A demonstra\u00e7\u00e3o que acabamos de revisar fundamenta apenas o caso em que as pot\u00eancias s\u00e3o n\u00fameros naturais, mas pode ser estendida para quaisquer n\u00fameros inteiros. Se <span class=\"katex-eq\" data-katex-display=\"false\">a\\in \\mathbb{Z}<\/span>, ent\u00e3o teremos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^a}{dx} = ax^{a-1}<\/span>\n<p>J\u00e1 sabemos que isso funciona para os inteiros positivos, s\u00f3 precisamos ver o que acontece quando tomamos pot\u00eancias negativas. \u00c9 suficiente, portanto, provar que se cumpre<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx^{-n}}{dx} = {-n}x^{-n-1}<\/span>\n<p><span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span> Para fazer isso, basta considerar a derivada do quociente<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}x^{-n} &amp;= \\dfrac{d}{dx} \\left( \\dfrac{1}{x^n}\\right) \\\\ \\\\\n\n&amp; &amp;= \\dfrac{0 \\cdot nx^{n-1} - nx^{n-1} \\cdot 1}{x^{2n}}\\\\ \\\\\n\n&amp; &amp;= -nx^{n-1-2n} \\\\ \\\\\n\n&amp; &amp;= -nx^{-n-1}\n\n\\end{array}\n\n<\/span>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Derivadas das fun\u00e7\u00f5es transcendentes<\/h2>\n<h3>Fun\u00e7\u00f5es trigonom\u00e9tricas<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=667s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Estas englobam as seguintes<\/span><\/strong><\/a> regras de deriva\u00e7\u00e3o:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sin(x) = \\cos(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\sec(x) = \\sec(x)\\tan(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cos(x) = -\\sin(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\csc(x) = -\\csc(x)\\cot(x)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\tan(x) = \\sec^2(x)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\cot(x) = -\\csc^2(x)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Para obter cada uma dessas regras, o melhor caminho \u00e9 come\u00e7ar pelas derivadas do seno e do cosseno; e depois, a partir desses resultados, utilizando a \u00e1lgebra das derivadas, obter as das demais fun\u00e7\u00f5es trigonom\u00e9tricas.<\/p>\n<h4>Demonstra\u00e7\u00e3o da derivada do seno<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp;\\dfrac{d}{dx}\\sin(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x+\\Delta x) - \\sin(x)}{\\Delta x} &amp; \\text{; Def. da derivada do seno} \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(x)\\cos(\\Delta x) + \\sin(\\Delta x)\\cos(x) - \\sin(x)}{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ \\sin(x)\\left[\\cos(\\Delta x) -1\\right] + \\sin(\\Delta x)\\cos(x) }{\\Delta x} &amp; \\\\ \\\\\n\n&amp;\\phantom{\\dfrac{d}{dx}\\sin(x)} = \\displaystyle \\sin(x)\\lim_{\\Delta x \\to 0} \\left[\\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\right] + \\cos(x) \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\right] &amp; \\\\ \\\\\n\n(2)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} = 1 &amp; \\text{; Pelo teorema do sandu\u00edche} \\\\ \\\\\n\n(3)&amp;\\displaystyle \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x} \\cdot \\dfrac{\\cos(\\Delta x) + 1}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{\\cos^2(\\Delta x) - 1}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} = \\lim_{\\Delta x\\to 0} \\dfrac{-\\sin^2(\\Delta x)}{\\Delta x (\\cos(\\Delta x) + 1)} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x} \\cdot \\lim_{\\Delta x\\to 0} \\dfrac{\\sin(\\Delta x)}{\\cos(\\Delta x) + 1} &amp; \\\\ \\\\\n\n&amp;\\displaystyle\\phantom{\\lim_{\\Delta x\\to 0} \\dfrac{\\cos(\\Delta x) - 1}{\\Delta x}} =- (1)\\cdot(0) = 0 \\\\ \\\\\n\n(4) &amp;\\color{blue}\\dfrac{d}{dx}\\sin(x) = \\cos(x) \\color{black} &amp; \\text{; De (1,2,3)}\n\n\\end{array}<\/span>\n<h4>Demonstra\u00e7\u00e3o da derivada do cosseno<\/h4>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx}\\cos(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x + \\Delta x) - \\cos(x)}{\\Delta x} &amp; \\text{; Def. da derivada do cosseno} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x)\\cos(\\Delta x) - \\sin(x)\\sin(\\Delta x) - \\cos(x)}{\\Delta x} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{\\cos(x) [ \\cos(\\Delta x) - 1] - \\sin(x)\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\displaystyle \\lim_{\\Delta x \\to 0} \\dfrac{ [ \\cos(\\Delta x) - 1]}{\\Delta x} - \\sin(x) \\lim_{\\Delta x \\to 0} \\dfrac{\\sin(\\Delta x)}{\\Delta x}\\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx}\\cos(x)} = \\cos(x) \\cdot(0) - \\sin(x)\\cdot (1)\\\\ \\\\\n\n&amp;\\color{blue}\\dfrac{d}{dx}\\cos(x) = - \\sin(x) \\color{black}\n\n\\end{array}<\/span>\n<h4>Derivadas da tangente, secante, cossecante e cotangente<\/h4>\n<p>Com os resultados para o seno e o cosseno, a obten\u00e7\u00e3o da derivada das demais fun\u00e7\u00f5es trigonom\u00e9tricas agora se torna simples.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\dfrac{d}{dx}\\tan(x) &amp;= \\dfrac{d}{dx} \\left( \\dfrac{\\sin(x)}{\\cos(x)} \\right) = \\dfrac{\\cos^2(x) + \\sin^2(x)}{\\cos^2(x)} = \\dfrac{1}{\\cos^2(x)} = \\color{blue}\\sec^2(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\sec(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\cos(x)} \\right) = \\dfrac{\\sin(x)}{\\cos^2(x)} =\\color{blue}\\sec(x)\\tan(x) \\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx}\\csc(x) &amp;= \\dfrac{d}{dx}\\left(\\dfrac{1}{\\sin(x)}\\right) = -\\dfrac{\\cos(x)}{\\sin^2(x)} =\\color{blue} - \\csc(x)\\cot(x)\\color{black}\\\\ \\\\\n\n\\dfrac{d}{dx} \\cot(x) &amp;= \\dfrac{d}{dx} \\left(\\dfrac{\\cos(x)}{\\sin(x)}\\right) = \\dfrac{-\\sin^2(x)-\\cos^2(x)}{\\sin^2(x)} = -\\dfrac{1}{\\sin^2(x)} =\\color{blue} -\\csc^2(x)\\color{black}\n\n\\end{array}<\/span>\n<h3>Derivada das fun\u00e7\u00f5es logar\u00edtmicas<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=1331s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A derivada do logaritmo natural<\/span><\/strong><\/a> \u00e9 dada por<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d}{dx}\\ln(x) = \\frac{1}{x}<\/span>\n<p><span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span> Raciocinando a partir da defini\u00e7\u00e3o da derivada, temos o seguinte desenvolvimento:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\dfrac{d}{dx} \\ln(x) = \\displaystyle \\lim_{\\Delta x \\to 0} \\left [\\dfrac{\\ln(x+\\Delta x) - \\ln(x)}{\\Delta x} \\right] &amp;\\text{; Def. da derivada para o Logaritmo Natural} \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{\\Delta x} \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right) \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\ln \\left( \\dfrac{x+\\Delta x}{x} \\right)^{\\frac{1}{\\color{red}x\\color{black}} \\frac{\\color{red}x\\color{black}}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} = \\displaystyle \\lim_{\\Delta x \\to 0} \\left[ \\dfrac{1}{x} \\ln \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n&amp; \\phantom{\\dfrac{d}{dx} \\ln(x)} =\\dfrac{1}{x} \\ln \\displaystyle \\left[ \\lim_{\\Delta x \\to 0} \\left( 1 + \\dfrac{\\Delta x}{x} \\right)^{ \\frac{x}{\\Delta x} } \\right] \\\\ \\\\\n\n(2) &amp; n=\\dfrac{x}{\\Delta x} &amp; \\text{; substitui\u00e7\u00e3o} \\\\ \\\\\n\n(3) &amp; (\\Delta x \\to 0^+) \\longrightarrow (n\\to +\\infty) \\\\ \\\\\n\n(4) &amp; \\dfrac{d}{dx} \\ln(x) = \\dfrac{1}{x} \\ln\\left[ \\displaystyle \\lim_{n \\to +\\infty} \\left(1 + \\dfrac{1}{n} \\right)^n \\right] = \\dfrac{1}{x} \\ln(e) = \\color{blue}\\dfrac{1}{x} \\color{black} &amp; \\text{; De (1,2,3)}\n\n\\end{array}<\/span>\n<p>\nCom isso, percorremos passo a passo as derivadas fundamentais que todo estudante deve dominar: desde fun\u00e7\u00f5es alg\u00e9bricas b\u00e1sicas at\u00e9 as principais fun\u00e7\u00f5es transcendentes, como as trigonom\u00e9tricas e o logaritmo natural. Dominando essas demonstra\u00e7\u00f5es, voc\u00ea poder\u00e1 aplicar as regras de deriva\u00e7\u00e3o, compreender sua origem e justificativa formal. Esse conhecimento \u00e9 a base para enfrentar com confian\u00e7a problemas mais complexos que exigem uma an\u00e1lise precisa da mudan\u00e7a.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Derivadas de polin\u00f4mios, trigonom\u00e9tricas e logaritmo A derivada \u00e9 uma ferramenta central do c\u00e1lculo diferencial, com aplica\u00e7\u00f5es fundamentais nas ci\u00eancias, na engenharia e na economia. Este artigo oferece um guia progressivo para dominar a diferencia\u00e7\u00e3o de fun\u00e7\u00f5es, desde polin\u00f4mios at\u00e9 fun\u00e7\u00f5es trigonom\u00e9tricas e logar\u00edtmicas. Por meio de demonstra\u00e7\u00f5es e exemplos concretos, busca-se compreender tanto a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":33155,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[856,571],"tags":[],"class_list":["post-33359","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Derivadas de polin\u00f4mios, fun\u00e7\u00f5es trigonom\u00e9tricas e logaritmos - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda passo a passo a derivar fun\u00e7\u00f5es alg\u00e9bricas, trigonom\u00e9tricas e logar\u00edtmicas com exemplos, demonstra\u00e7\u00f5es claras e t\u00e9cnicas eficazes.\" \/>\n<meta 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