{"id":182,"date":"2019-08-27T17:48:19","date_gmt":"2019-08-27T17:48:19","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=182"},"modified":"2020-06-17T19:01:13","modified_gmt":"2020-06-17T19:01:13","slug":"limit-definition-of-the-derivative","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/limit-definition-of-the-derivative\/","title":{"rendered":"Limit Definition of the Derivative"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Limit Definition of the Derivative &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nOnce we know the most basic differentiation formulas and rules, we\ncompute new derivatives using what we already know.  We rarely think\nback to where the basic formulas and rules originated.\n\n<\/p>\n\n\n\n<p>\nThe geometric meaning of the derivative\n\\[f'(x)=\\frac{df(x)}{dx}\\]\nis the slope of the line tangent to $y=f(x)$ at $x$.\n\n<\/p>\n\n\n\n<p>\n\nLet&#8217;s look for this slope at $P$:\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"239\" height=\"162\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/lim_figure1.gif?resize=239%2C162&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-357\"\/><\/figure><\/div>\n\n\n\n<p>\nThe <b>secant<\/b> line through $P$ and $Q$ has slope\n\\[\\frac{f(x+\\Delta x)-f(x)}{(x+\\Delta x)-x}=\\frac{f(x+\\Delta x)-f(x)}{\\Delta x}.\\]\n\n<\/p>\n\n\n\n<p>\nWe can approximate the <b>tangent<\/b> line through $P$ by moving $Q$\ntowards $P$, decreasing $\\Delta x$.  In the limit as $\\Delta x\\to 0$, we get the\ntangent line through $P$ with slope \n\\[\\lim_{\\Delta x\\to 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}.\\]\nWe define\n\\[f'(x)=\\lim_{\\Delta x\\to 0}\\frac{f(x+\\Delta x)-f(x)^{\\small\\textrm{*}}}{\\Delta x}.\\]\n\n<\/p>\n\n\n\n<p>\n$^*$ If the limit as $\\Delta x \\to 0$ at a particular point does not exist, \n$f'(x)$ is undefined at that point.\n\n<\/p>\n\n\n\n<p>\nWe derive all the basic differentiation formulas using this\ndefinition.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nFor $f(x)=x^2$,\n\\begin{eqnarray*}\n\tf'(x)&amp;=&amp;\\lim_{\\Delta x\\to 0} \\frac{(x+\\Delta x)^2-x^2}{\\Delta x}\\\\\n\t&amp;=&amp; \\lim_{\\Delta x\\to 0} \\frac{(x^2+2(\\Delta x)x+\\Delta x^2)-x^2}{\\Delta x}\\\\\n\t&amp;=&amp;\\lim_{\\Delta x\\to 0} \\frac{2(\\Delta x)x+\\Delta x^2}{\\Delta x}\\\\\n\t&amp;=&amp;\\lim_{\\Delta x\\to 0} (2x+\\Delta x)\\\\\n\t&amp;=&amp; 2x \n\\end{eqnarray*}\nas expected.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nFor $\\displaystyle f(x)=\\frac{1}{x}$\n\\begin{eqnarray*}\n\tf'(x)&amp;=&amp;\\lim_{\\Delta x\\to 0}\\frac{\\frac{1}{x+\\Delta x}-\\frac{1}{x}}{\\Delta x} \\\\\n\t&amp;=&amp;\\lim_{\\Delta x\\to 0}\\frac{\\frac{x-(x+\\Delta x)}{(x+\\Delta x)(x)}}{\\Delta x} \\\\\n\t&amp;=&amp;\\lim_{\\Delta x\\to 0}\\frac{\\frac{-\\Delta x}{(x+\\Delta x)(x)}}{\\Delta x} \\\\\n\t&amp;=&amp;\\lim_{\\Delta x\\to 0} \\frac{-1}{(x+\\Delta x)(x)}\\\\\n\t&amp;=&amp; -\\frac{1}{x^2}\n\\end{eqnarray*}\nagain as expected.\n\n<\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<p>\n\nThe limit definition of the derivative is used to prove many\nwell-known results, including the following:\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> If $f$ is differentiable at $x_0$, then $f$ is continuous at\n\t\t$x_0$.\n\t<\/li><li> Differentiation of polynomials: $\\displaystyle\n\t\t\\frac{d}{dx}\\left[x^n\\right]=nx^{n-1}$.\n\t<\/li><li> Product and Quotient Rules for differentiation.\n<\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<h4>Key Concepts<\/h4>\n<\/center>\n\n\n\n<p>\n\nWe define\n$f'(x) = \\displaystyle\\lim_{\\Delta x\\to 0}\\frac{f(x+\\Delta x)-f(x)}{\\Delta x}$.\n\n<br><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1810\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Limit Definition of the Derivative &#8211; HMC Calculus Tutorial Once we know the most basic differentiation formulas and rules, we compute new derivatives using what we already know. We rarely think back to where the basic formulas and rules originated. The geometric meaning of the derivative \\[f'(x)=\\frac{df(x)}{dx}\\] is the slope of the line tangent to&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-182","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=182"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/182\/revisions"}],"predecessor-version":[{"id":1214,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/182\/revisions\/1214"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=182"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}