{"id":29707,"date":"2024-11-27T12:00:27","date_gmt":"2024-11-27T12:00:27","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29707"},"modified":"2024-11-27T17:35:39","modified_gmt":"2024-11-27T17:35:39","slug":"the-derivative-as-the-limit-of-a-function","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-derivative-as-the-limit-of-a-function\/","title":{"rendered":"The Derivative as the Limit of a Function"},"content":{"rendered":"<style>\np {\n  text-align: justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">The Derivative as the Limit of a Function<\/h1>\n<p style=\"text-align:center;\">\n  <em><strong>Abstract:<\/strong> In this lesson, we will explore the concept of the derivative as the mathematical tool to analyze changes in functions. Starting from the slope of a secant line, and taking the limit as the points get closer, we will define the derivative as the slope of the tangent line. Additionally, we will study its key properties and rules, such as sum, product, and quotient rules, which are fundamental for applying derivatives in the analysis of functions and phenomena of change.<\/em>\n<\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives<\/strong><\/p>\n<p>By the end of this lesson, students will be able to:<\/p>\n<ol>\n<li><strong>Understand<\/strong> the derivative as the limit describing the instantaneous change in a function and as the slope of the tangent line to a curve at a point.<\/li>\n<li><strong>Explain<\/strong> how differentiability implies continuity in functions.<\/li>\n<li><strong>Demonstrate<\/strong> the basic differentiation rules from the formal definition.<\/li>\n<li><strong>Apply<\/strong> the properties of the algebra of derivatives (sum, product, and quotient) to mathematical problems.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE OF CONTENTS<\/u>:<\/strong><\/p>\n<p><a href=\"#1\"><strong>The concept of the derivative<\/strong><\/a><br \/>\n<a href=\"#1\">The slope of the secant line<\/a><br \/>\n<a href=\"#1\">Taking the limit: The derivative and the slope of the tangent line<\/a><br \/>\n<a href=\"#1\">Alternative definition<\/a><br \/>\n<a href=\"#1\"><strong>Properties of Derivatives<\/strong><\/a><br \/>\n<a href=\"#1\">Differentiability implies continuity<\/a><br \/>\n<a href=\"#1\">Algebra of derivatives<\/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>The Concept of the Derivative<\/h2>\n<p>Nature is generally subject to change, and the quintessential mathematical tool for calculating and understanding change is the derivative. This arises from asking, \u00abWhat happens to the value of a function <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> when the variable <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> is increased or decreased by an arbitrarily small amount <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span>?\u00bb The concept of the derivative emerges as the limit of a function when analyzing that question.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>The Slope of the Secant Line<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=164s\" target=\"_blank\" rel=\"noopener\"><strong>Consider a function<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> evaluated at two points <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. Any line cutting through two points of a curve is called a \u00absecant line\u00bb and looks like the one in the figure below.<\/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=\"Secant Line Graph\" 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=\"Secant Line Graph\" class=\"aligncenter lazyload\" width=\"397\" height=\"233\" \/><\/noscript><\/p>\n<p>The slope of this secant line is given by:<\/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>Taking the Limit: The Derivative and the Slope of the Tangent Line<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=278s\" target=\"_blank\" rel=\"noopener\"><strong>Consider the secant line of the curve<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x)<\/span> passing through <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">x_0 + \\Delta x<\/span>. If we take the limit as <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x<\/span> approaches zero, we obtain the tangent line to the curve at <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=\"Tangent Line Graph\" 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=\"Tangent Line Graph\" class=\"aligncenter lazyload\" width=\"464\" height=\"268\" \/><\/noscript><\/p>\n<p>From this, the formal definition of the derivative of a function <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> at a point <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> is given as:<\/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>This represents the slope of the tangent line passing through <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Alternative Definition<\/h3>\n<p>An alternative way of presenting the definition of the derivative as a limit is derived from the following substitution:<\/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>Thus, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta x = x_f - x_i<\/span>, and the definition of the derivative becomes:<\/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=\"Definition of derivative as the limit of secant slopes\" 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=\"Definition of derivative as the limit of secant slopes\" class=\"aligncenter lazyload\" width=\"469\" height=\"243\" \/><\/noscript><\/p>\n<p>Both definitions are equivalent and can be used interchangeably, depending on convenience.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Properties of Derivatives<\/h2>\n<p>A function is said to be differentiable at <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> if the following limit exists:<\/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>We say it is differentiable on a set <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> if the limit is well-defined for all <span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I.<\/span> Differentiable functions have the following properties:<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Differentiability Implies Continuity<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=526s\" target=\"_blank\" rel=\"noopener\"><strong>If a function is differentiable at <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">x_0,<\/span> then it is continuous at <span class=\"katex-eq\" data-katex-display=\"false\">x_0.<\/span> This can be proven through the following argument:<\/p>\n<p>For <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> to be continuous at <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, it is necessary that:<\/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>Examining the left-hand side of this expression, we have:<\/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>Therefore, for <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> to be continuous at <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, it is necessary that the right-hand limit is well-defined. This occurs if and only if:<\/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>In other words, if <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> is differentiable at <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>. Consequently, if <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> is differentiable at <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>, it is also continuous at that point.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Algebra of Derivatives<\/h3>\n<p>Let <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> be differentiable functions for all <span class=\"katex-eq\" data-katex-display=\"false\">x \\in I<\/span>, and let <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta \\in \\mathbb{R}.<\/span> Then, the following holds:<\/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>If <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0<\/span>, then <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>As we can see, the algebra of derivatives is not as intuitive as it might seem at first glance; however, these properties can be derived without much difficulty from the definition of derivatives as limits.<\/p>\n<p><span style=\"color: #000080;\">PROOF:<\/span><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=925s\" target=\"_blank\" rel=\"noopener\"><strong>The proof for the derivative of the sum<\/strong><\/a> follows this reasoning:<\/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>On the other hand, the proof for the derivative of the product<\/strong><\/a> is slightly more complex but manageable:<\/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>Here, we utilized the fact that if <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> is a differentiable function, it is also continuous. Thus, <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{\\Delta x \\to 0 } g(x+\\Delta x) = g(x)<\/span>. This conclusion is reached using <strong>limit algebra<\/strong>.<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=TFxATgmYvkY&amp;t=162s\" target=\"_blank\" rel=\"noopener\"><strong>Finally, for the proof of the derivative of the quotient,<\/strong><\/a> we can leverage the product rule. Consider a function of the form <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = f(x)\/g(x),<\/span> where <span class=\"katex-eq\" data-katex-display=\"false\">g(x) \\neq 0.<\/span> From this, we have:<\/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>Solving for <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{dk(x)}{dx}<\/span>, we find:<\/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>Thus:<\/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>This is what we aimed to prove.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Derivative as the Limit of a Function Abstract: In this lesson, we will explore the concept of the derivative as the mathematical tool to analyze changes in functions. Starting from the slope of a secant line, and taking the limit as the points get closer, we will define the derivative as the slope of [&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":7,"footnotes":""},"categories":[854,567],"tags":[],"class_list":["post-29707","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-differential-calculus","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The Derivative as the Limit of a Function - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"I explain the concept of the derivative as a limit of a function and teach you clearly and simply how to deduce all its properties.\" \/>\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\/en\/the-derivative-as-the-limit-of-a-function\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Derivative as the Limit of a Function\" \/>\n<meta property=\"og:description\" content=\"I explain the concept of the derivative as a limit of a function and teach you clearly and simply how to deduce all its properties.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/the-derivative-as-the-limit-of-a-function\/\" \/>\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=\"2024-11-27T12:00:27+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-11-27T17:35:39+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=\"The Derivative as the Limit of a Function\" \/>\n<meta name=\"twitter:description\" content=\"I explain the concept of the derivative as a limit of a function and teach you clearly and simply how to deduce all its properties.\" \/>\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=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Derivative as the Limit of a Function\",\"datePublished\":\"2024-11-27T12:00:27+00:00\",\"dateModified\":\"2024-11-27T17:35:39+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/\"},\"wordCount\":1655,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/11\\\/derivada014.jpg\",\"articleSection\":[\"Differential Calculus\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-derivative-as-the-limit-of-a-function\\\/\",\"name\":\"The Derivative as the Limit of a Function - 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