{"id":33017,"date":"2024-11-30T13:00:51","date_gmt":"2024-11-30T13:00:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=33017"},"modified":"2025-07-18T00:23:27","modified_gmt":"2025-07-18T00:23:27","slug":"derivadas-de-polinomios-trigonometricas-y-logaritmo","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/es\/derivadas-de-polinomios-trigonometricas-y-logaritmo\/","title":{"rendered":"Derivadas de polinomios, trigonom\u00e9tricas y logaritmo"},"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 polinomios, trigonom\u00e9tricas y logaritmo<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\nLa derivada es una herramienta central del c\u00e1lculo diferencial, con aplicaciones fundamentales en las ciencias, la ingenier\u00eda y la econom\u00eda. Este art\u00edculo ofrece una gu\u00eda progresiva para dominar la derivaci\u00f3n de funciones, desde polinomios hasta funciones trigonom\u00e9tricas y logar\u00edtmicas. A trav\u00e9s de demostraciones y ejemplos concretos, se busca comprender tanto la aplicaci\u00f3n de las reglas como su fundamento.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objetivos de Aprendizaje<\/strong><\/p>\n<ol>\n<li><strong>Comprender<\/strong> el concepto general de derivada y sus propiedades fundamentales.<\/li>\n<li><strong>Aplicar<\/strong> la definici\u00f3n formal de derivada para calcular derivadas b\u00e1sicas.<\/li>\n<li><strong>Demostrar<\/strong> mediante l\u00edmites la derivada de funciones constantes y funciones identidad.<\/li>\n<li><strong>Obtener<\/strong> las reglas para derivar funciones trigonom\u00e9tricas partiendo de las derivadas fundamentales del seno y coseno.<\/li>\n<li><strong>Calcular<\/strong> derivadas de funciones trigonom\u00e9tricas compuestas utilizando reglas algebraicas.<\/li>\n<li><strong>Demostrar<\/strong> formalmente la derivada del logaritmo natural mediante l\u00edmites.<\/li>\n<\/ol>\n<p style=\"text-align:center\">\n<strong><u>\u00cdNDICE DE CONTENIDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">Derivada de las funciones algebraicas<\/a><br \/>\n<a href=\"#2\">Derivadas de las funciones trascendentales<\/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>Hasta ahora s\u00f3lo hemos revisado qu\u00e9 es la derivada y algunas de sus propiedades algebraicas, pero nada hemos dicho sobre c\u00f3mo se calculan. Aqu\u00ed resolveremos ese problema mostrando cada una de las t\u00e9cnicas de derivaci\u00f3n y como se obtienen para cada caso.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Derivada de las funciones algebraicas<\/h2>\n<h3>Funci\u00f3n 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;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = c,<\/span> siendo <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> <\/strong><\/a>una constante real cualquiera, entonces se tendr\u00e1 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>DEMOSTRACI\u00d3N:<\/strong><\/span> En realidad esta demostraci\u00f3n se hace en un solo paso:<\/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. de 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>Funci\u00f3n identidad<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=169s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x,<\/span> entonces:<\/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>DEMOSTRACI\u00d3N:<\/strong><\/span> Casi igual al anterior, tambi\u00e9n sale en un solo paso:<\/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. de 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>Potencias naturales<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=LudVXLRnmLw&amp;t=239s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x^n,<\/span> donde<\/span> <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> es un natural cualquiera, entonces se tendr\u00e1 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>DEMOSTRACI\u00d3N:<\/strong><\/span> Para demostrar este teorema, tendremos que usar el <a href=\"https:\/\/toposuranos.com\/ejercicios-de-induccion-matematica-teorema-del-binomio-de-newton\/\" target=\"_blank\" rel=\"noopener\">teorema del binomio 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 l\u00edmite 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 del Binomio 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 el primer t\u00e9rmino de la suma} \\\\ \\\\\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 t\u00e9rminos semejantes} \\\\ \\\\\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{; Extrayendo el primer t\u00e9rmino de la suma} \\\\ \\\\\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>Potencias Enteras<\/h3>\n<p>La demostraci\u00f3n que se acaba de revisar s\u00f3lo fundamenta el caso en que las potencias son n\u00fameros naturales, pero se puede extender para n\u00fameros enteros cualesquiera. Si <span class=\"katex-eq\" data-katex-display=\"false\">a\\in \\mathbb{Z}<\/span>, entonces se tendr\u00e1 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>Ya sabemos que esto funciona para los enteros positivos, s\u00f3lo nos basta ver lo que pasa cuando tomamos potencias negativas. Es suficiente con probar, por lo tanto, que se cumple<\/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>DEMOSTRACI\u00d3N:<\/strong><\/span> Para hacer esto basta con considerar la derivada del cociente<\/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 de las funciones trascendentales<\/h2>\n<h3>Funciones 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 engloban las siguiente<\/span><\/strong><\/a> reglas de derivaci\u00f3n:<\/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 obtener cada una de estas reglas, el mejor camino es partir por las derivadas del seno y del coseno; y luego, a partir de estos resultados, utilizando el \u00e1lgebra de las derivadas obtener la de las dem\u00e1s trigonom\u00e9tricas.<\/p>\n<h4>Demostraci\u00f3n de la derivada del 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. de derivada del 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{; Por el teorema del Sandwich}\\\\ \\\\\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>Demostraci\u00f3n de la derivada del coseno<\/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. de derivada del coseno} \\\\ \\\\\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 de la tangente, secante, cosecante y cotangente<\/h4>\n<p>Teniendo los resultados para el seno y el coseno, la obtenci\u00f3n de la derivada para las dem\u00e1s trigonom\u00e9tricas ahora son f\u00e1ciles de obtener.<\/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}{\\cos(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 de las funciones 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;\">La derivada del logaritmo natural<\/span><\/strong><\/a> est\u00e1 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>DEMOSTRACI\u00d3N:<\/strong><\/span> Razonando desde la definici\u00f3n de la derivada se tiene el siguiente razonamiento:<\/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. de derivada para 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{; sustituci\u00f3n}\\\\ \\\\\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>\nCon esto hemos recorrido paso a paso las derivadas fundamentales que todo estudiante debe dominar: desde funciones algebraicas b\u00e1sicas hasta las principales funciones trascendentales como las trigonom\u00e9tricas y el logaritmo natural. Dominando estas demostraciones podr\u00e1s aplicar las reglas de derivaci\u00f3n, comprender su origen y justificaci\u00f3n formal. Este conocimiento es la base para abordar con confianza problemas m\u00e1s complejos que requieren un an\u00e1lisis preciso del cambio.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Derivadas de polinomios, trigonom\u00e9tricas y logaritmo La derivada es una herramienta central del c\u00e1lculo diferencial, con aplicaciones fundamentales en las ciencias, la ingenier\u00eda y la econom\u00eda. Este art\u00edculo ofrece una gu\u00eda progresiva para dominar la derivaci\u00f3n de funciones, desde polinomios hasta funciones trigonom\u00e9tricas y logar\u00edtmicas. A trav\u00e9s de demostraciones y ejemplos concretos, se busca comprender [&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":50,"footnotes":""},"categories":[852,563],"tags":[],"class_list":["post-33017","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculo-diferencial","category-matematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Derivadas de polinomios, trigonom\u00e9tricas y logaritmo - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprende paso a paso a derivar funciones algebraicas, trigonom\u00e9tricas y logar\u00edtmicas con ejemplos, demostraciones claras y t\u00e9cnicas efectivas.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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