{"id":34342,"date":"2024-11-27T12:00:57","date_gmt":"2024-11-27T12:00:57","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34342"},"modified":"2025-09-07T03:18:46","modified_gmt":"2025-09-07T03:18:46","slug":"derivata-ut-limes-functionis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/","title":{"rendered":"Derivata ut Limes Functionis"},"content":{"rendered":"<style>\np{\ntext-align:justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">Derivata ut Limes Functionis<\/h2>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione explorabimus notionem derivatae ut instrumentum mathematicum ad mutationes in functionibus investigandas. Incipiemus a declivitate lineae secantis et, capto limite cum puncta appropinquant, definibimus derivatam ut declivitatem tangenti. Praeterea proprietates eius praecipuas atque regulas, ut additionis, multiplicationis et divisionis, investigabimus, fundamentales ad applicationem derivatarum in analysi functionum et phaenomenorum mutationis.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi<\/strong><br \/>\nAd finem huius lectionis, studiosus poterit:\n<\/p>\n<ol>\n<li><strong>Intellegere<\/strong> derivatam ut limitem qui mutationem instantaneam in functione describit atque ut declivitatem lineae tangentis ad curvam in puncto.<\/li>\n<li><strong>Explicare<\/strong> quomodo derivabilitas implicationem habeat continuitatis in functionibus.<\/li>\n<li><strong>Demonstr\u0101re<\/strong> regulas fundamentales derivandi ex definitione formali.<\/li>\n<li><strong>Adhibere<\/strong> proprietates algebrae derivatarum (additio, multiplicatio et divisio) in problematibus mathematicis.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>Notio derivatae<\/strong><\/a><br \/>\n<a href=\"#2\">Declivitas lineae secantis<\/a><br \/>\n<a href=\"#3\">Transitus ad limitem: Derivata et declivitas lineae tangentis<\/a><br \/>\n<a href=\"#4\">Definitio alternativa<\/a><br \/>\n<a href=\"#5\"><strong>Proprietates Derivatarum<\/strong><\/a><br \/>\n<a href=\"#6\">Derivabilitas implicat continuitatem<\/a><br \/>\n<a href=\"#\">Algebra derivatarum<\/a>\n<\/p>\n<p><center><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><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Notio derivatae<\/h2>\n<p>Natura in universum est susceptibilis mutationis, et instrumentum mathematicum praecipuum ad mutationem computandam atque intellegendam est Derivata. Haec oritur ex interrogatione \u00abquid fiet cum valore functionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> cum variabilis <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> augetur vel minuitur quantitate tam parva quam volumus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span><\/span>?\u00bb. Notio derivatae emergit ut limes functionis dum haec quaestio investigatur.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Declivitas lineae secantis<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=164s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremus functionem<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> in duobus punctis aestimatam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span><\/span>. Omnis linea quae duos punctos curva intersecat appellatur \u00ablinea secans\u00bb et apparet sicut in imagine monstratur.<\/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=\"Grafico recta 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=\"Grafico recta secante\" class=\" aligncenter lazyload\" width=\"397\" height=\"233\" \/><\/noscript><\/p>\n<p>Haec linea secans in specie habet declivitatem<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><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><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Transitus ad limitem: Derivata et declivitas lineae tangentis<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=278s\" target=\"_blank\" rel=\"noopener\"><strong>Si consideremus lineam secantem curvae<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span><\/span> quae transit per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span><\/span>, deinde capiamus limitem cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span><\/span> tendit ad nihilum, quod obtinebimus est linea tangens curvae quae transit per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(x_0, f(x_0)).<\/span><\/span><\/p>\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 recta 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 recta tangente\" class=\" aligncenter lazyload\" width=\"464\" height=\"268\" \/><\/noscript><\/p>\n<p>Ex hoc emergit definitio formalis derivatae functionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> in puncto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> ut limes<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><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><\/span><\/p>\n<p>quae simul repraesentat declivitatem lineae tangentis quae transit per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Definitio alternativa<\/h3>\n<p>Modus alternativus exhibendi definitionem derivatae ut limitem obtinetur ex hac substitutione:<\/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>Hoc habebimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_f - x_i<\/span><\/span> et definitio derivatae se habebit hoc modo<\/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=\"definici\u00f3n de derivada como el l\u00edmite de las pendientes de las rectas 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=\"definici\u00f3n de derivada como el l\u00edmite de las pendientes de las rectas secantes\" class=\" aligncenter lazyload\" width=\"469\" height=\"243\" \/><\/noscript><\/p>\n<p>Utraque definitio aequipollens est et adhiberi potest alternatim pro opportunitate.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Proprietates Derivatarum<\/h2>\n<p>Dicitur functio derivabilis esse in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> cum existat limes<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><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><\/span><\/p>\n<p>Et dicemus eam derivabilem esse in coniunctione <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> si limes bene definitus est pro omnibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I.<\/span><\/span> Functiones derivabiles has proprietates habent:<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Derivabilitas implicat continuitatem<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=526s\" target=\"_blank\" rel=\"noopener\"><strong>Si functio derivabilis est in <\/strong><\/a><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, tunc continua est in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Hoc demonstrare possumus per sequentem rationem.<\/p>\n<p>Ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> sit continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> necesse est ut impleatur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = f(x_0) <\/span><\/span><\/p>\n<p>Si latus sinistrum huius expressionis examinemus habebimus:<\/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\n\\end{array}\n\n<\/span>\n<p>Hinc habetur ut, ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> sit continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, necesse sit ut limes dexter bene definitus sit; quod fit si et solum si<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><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><\/span><\/p>\n<p>Aliter dicendo, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> derivabilis est in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Consequenter, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> derivabilis est in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, tunc continua est in illo puncto.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Algebra derivatarum<\/h3>\n<p>Sint <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> functiones derivabiles pro omnibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in I<\/span><\/span>, et sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta\\in\\mathbb{R}.<\/span><\/span> Tunc habetur:<\/p>\n<ol>\n<li><span dir=\"ltr\"><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><\/span><\/li>\n<\/li>\n<li><span dir=\"ltr\"><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><\/span><\/li>\n<li>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)\\neq 0<\/span><\/span>, tunc <span dir=\"ltr\"><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><\/span><\/li>\n<\/ol>\n<p>Ut videre possumus, algebra derivatarum non est tam intuitiva quam primo aspectu videri potest; tamen demonstratio harum proprietatum sine magna difficultate ex definitione derivatarum ut limitum deduci potest.<\/p>\n<p><span style=\"color: #000080;\">DEMONSTRATIO:<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=925s\" target=\"_blank\" rel=\"noopener\"><strong>Demonstratio derivatae additionis<\/strong><\/a> nobis relinquitur hoc ratiocinio sequendo:<\/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 style=\"text-align: center;\">\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=1059s\" target=\"_blank\" rel=\"noopener\"><strong>Altero modo, demonstratio derivatae producti<\/strong><\/a> paulo difficilior est, sed nihil extraordinarium:<\/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>Hic adhibitum est quod, cum <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> functio derivabilis sit, tunc continua est atque ideo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{\\Delta x\\to 0 } g(x+\\Delta x) = g(x)<\/span><\/span>, ut postea conclusa sit demonstratio utens <strong>algebra limitum.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=162s\" target=\"_blank\" rel=\"noopener\"><strong>Denique, ad demonstrationem derivatae divisionis,<\/strong><\/a> possumus uti effectu derivatae multiplicationis. Consideremus functionem huius formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k(x) = f(x)\/g(x)<\/span><\/span>, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)\\neq 0<\/span><\/span>. Ex hoc habebitur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><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><\/span><\/p>\n<p>Nunc, solvendo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}<\/span><\/span> habetur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><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><\/span><\/p>\n<p>Atque ideo:<\/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>quod erat demonstrandum.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Derivata ut Limes Functionis Summarium: In hac lectione explorabimus notionem derivatae ut instrumentum mathematicum ad mutationes in functionibus investigandas. Incipiemus a declivitate lineae secantis et, capto limite cum puncta appropinquant, definibimus derivatam ut declivitatem tangenti. Praeterea proprietates eius praecipuas atque regulas, ut additionis, multiplicationis et divisionis, investigabimus, fundamentales ad applicationem derivatarum in analysi functionum et [&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":93,"footnotes":""},"categories":[1328,1298],"tags":[],"class_list":["post-34342","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-differentialis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Derivata ut Limes Functionis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Tibi explico notionem derivatae ut limitem functionis, atque doceo clare et simpliciter quomodo omnes eius proprietates deducere possis.\" \/>\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\/la\/derivata-ut-limes-functionis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivata ut Limes Functionis\" \/>\n<meta property=\"og:description\" content=\"Tibi explico notionem derivatae ut limitem functionis, atque doceo clare et simpliciter quomodo omnes eius proprietates deducere possis.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/\" \/>\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=\"2024-11-27T12:00:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-07T03:18:46+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=\"Derivata ut Limes Functionis\" \/>\n<meta name=\"twitter:description\" content=\"Tibi explico notionem derivatae ut limitem functionis, atque doceo clare et simpliciter quomodo omnes eius proprietates deducere possis.\" \/>\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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Derivata ut Limes Functionis\",\"datePublished\":\"2024-11-27T12:00:57+00:00\",\"dateModified\":\"2025-09-07T03:18:46+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/\"},\"wordCount\":1486,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/derivada014.jpg\",\"articleSection\":[\"Calculus Differentialis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/\",\"url\":\"http:\/\/toposuranos.com\/material\/la\/derivata-ut-limes-functionis\/\",\"name\":\"Derivata ut Limes Functionis - 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