{"id":1616,"date":"2025-07-25T04:07:49","date_gmt":"2025-07-25T04:07:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1616"},"modified":"2026-03-25T05:24:32","modified_gmt":"2026-03-25T05:24:32","slug":"derivatives-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/derivatives-fresh-take\/","title":{"raw":"Derivatives: Fresh Take","rendered":"Derivatives: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the derivative of a function.<\/li>\r\n \t<li>Find instantaneous rates of change.<\/li>\r\n \t<li>Find an equation of the tangent line to the graph of a function at a point.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivative Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Derivative Function:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Gives the derivative of a function at every point in its domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Defined as: [latex]f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Differentiability:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A function is differentiable at a point if its derivative exists at that point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">A function is differentiable on an interval if it's differentiable at every point in that interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Notation for Derivatives:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f'(x)[\/latex], [latex]y'[\/latex], [latex]\\frac{dy}{dx}[\/latex], [latex]\\frac{d}{dx}(f(x))[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Leibniz Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represents instantaneous rate of change<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169737819520\">Find the derivative of [latex]f(x)=x^2[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737789025\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737789025\"]\r\n<p id=\"fs-id1169737789025\">[latex]f^{\\prime}(x)=2x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to this example.<center><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bcehahhc-w9m4h4i47-M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/w9m4h4i47-M?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bcehahhc-w9m4h4i47-M\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6246501&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bcehahhc-w9m4h4i47-M&vembed=0&video_id=w9m4h4i47-M&video_target=tpm-plugin-bcehahhc-w9m4h4i47-M'><\/script><\/p>\r\n<p style=\"text-align: left;\">Watch the clip from 5:49 until 6:48. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\r\n<p style=\"text-align: left;\">You can view the\u00a0<a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.2TheDerivativeAsAFunction349to408_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.2 The Derivative as a Function\" here (opens in new window).<\/a> If you would like to watch the entire video, you can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/3.2+The+Derivative+as+a+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c3.2 The Derivative as a Function\u201d here (opens in new window).<\/a><\/p>\r\n\r\n<\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">Find the derivative of [latex]f(x) = x^3 + 2x[\/latex].[reveal-answer q=\"255451\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"255451\"]\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} f'(x) &amp;=&amp; \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} \\frac{((x+h)^3 + 2(x+h)) - (x^3 + 2x)}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} \\frac{(x^3 + 3x^2h + 3xh^2 + h^3 + 2x + 2h) - (x^3 + 2x)}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} \\frac{3x^2h + 3xh^2 + h^3 + 2h}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} (3x^2 + 3xh + h^2 + 2) \\\\ &amp;=&amp; 3x^2 + 2 \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]f'(x) = 3x^2 + 2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-badcaeeh-N2PpRnFqnqY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/N2PpRnFqnqY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-badcaeeh-N2PpRnFqnqY\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661480&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-badcaeeh-N2PpRnFqnqY&vembed=0&video_id=N2PpRnFqnqY&video_target=tpm-plugin-badcaeeh-N2PpRnFqnqY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Derivative+as+a+concept+%7C+Derivatives+introduction+%7C+AP+Calculus+AB+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDerivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section aria-label=\"Watch It\">\r\n<h2>The Basic Rules<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Constant Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For any constant [latex]c[\/latex], [latex]\\frac{d}{dx}(c) = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Power Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]f(x) = x^n[\/latex] where [latex]n[\/latex] is a positive integer: [latex]\\frac{d}{dx}(x^n) = nx^{n-1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sum and Difference Rules:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Constant Multiple Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For any constant [latex]k[\/latex], [latex]\\frac{d}{dx}(kf(x)) = kf'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">These rules form the foundation for differentiating more complex functions.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The Power Rule applies to positive integer exponents and will be extended to other exponents later.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These rules allow us to differentiate polynomials and many other functions without using the limit definition every time.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736614166\">Find the derivative of [latex]g(x)=-3[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738853102\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738853102\"]\r\n<p id=\"fs-id1169738853102\">[latex]0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169738916812\">Find [latex]\\frac{d}{dx}(x^4)[\/latex]<\/p>\r\n[reveal-answer q=\"41137798\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"41137798\"]\r\n<p id=\"fs-id1169739270315\">Use [latex](x+h)^4=x^4+4x^3h+6x^2h^2+4xh^3+h^4[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739000891\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739000891\"]\r\n<p id=\"fs-id1169739000891\">[latex]4x^3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739199747\">Find the derivative of [latex]f(x)=x^7[\/latex].<\/p>\r\n[reveal-answer q=\"25547709\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"25547709\"]\r\n<p id=\"fs-id1169736613524\">Use the power rule with [latex]n=7[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169738962015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738962015\"]\r\n<p id=\"fs-id1169738962015\">[latex]f^{\\prime}(x)=7x^6[\/latex]<\/p>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\" style=\"text-align: center;\"><iframe id=\"tpm-plugin-aeabafae-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aeabafae-ruACLHzWT3g\" class=\"p3sdk-target\" style=\"text-align: center;\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6246502&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aeabafae-ruACLHzWT3g&vembed=0&video_id=ruACLHzWT3g&video_target=tpm-plugin-aeabafae-ruACLHzWT3g'><\/script><\/p>\r\nWatch the clip from 3:01 until 3:11. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules181to191_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>. You can also access the entire <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/3.3+Differentiation+Rules_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c3.3 Differentiation Rules\u201d here (opens in new window).<\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739299469\">Find the derivative of [latex]f(x)=2x^3-6x^2+3[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736658726\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658726\"]\r\n<p id=\"fs-id1169736658726\">[latex]f^{\\prime}(x)=6x^2-12x[\/latex].<\/p>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\" style=\"text-align: center;\"><iframe id=\"tpm-plugin-dfbdgbbc-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dfbdgbbc-ruACLHzWT3g\" class=\"p3sdk-target\" style=\"text-align: center;\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6246502&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dfbdgbbc-ruACLHzWT3g&vembed=0&video_id=ruACLHzWT3g&video_target=tpm-plugin-dfbdgbbc-ruACLHzWT3g'><\/script><\/p>\r\nWatch the clip from 5:00 until 5:30. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules300to330_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169736654291\">Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[\/latex] at [latex]x=2[\/latex]. Use the point-slope form.<\/p>\r\n[reveal-answer q=\"fs-id1169739353706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739353706\"]\r\n<p id=\"fs-id1169739353706\">[latex]y=12x-23[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 class=\"entry-title\">The Advanced Rules<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Product Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]j(x) = f(x)g(x)[\/latex]: [latex]j'(x) = f'(x)g(x) + g'(x)f(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quotient Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]j(x) = \\frac{f(x)}{g(x)}[\/latex]: [latex]j'(x) = \\frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Extended Power Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]k[\/latex] a negative integer: [latex]\\frac{d}{dx}(x^k) = kx^{k-1}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The product rule is not simply the product of the derivatives.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The quotient rule involves a specific arrangement of terms in the numerator.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The extended power rule allows differentiation of negative integer powers.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These rules expand our ability to differentiate more complex functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736654828\">Use the product rule to obtain the derivative of [latex]j(x)=2x^5(4x^2+x)[\/latex].<\/p>\r\n[reveal-answer q=\"034256\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"034256\"]\r\n<p id=\"fs-id1169736609959\">Set [latex]f(x)=2x^5[\/latex] and [latex]g(x)=4x^2+x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736654876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736654876\"]\r\n<p id=\"fs-id1169736654876\">[latex]j^{\\prime}(x)=10x^4(4x^2+x)+(8x+1)(2x^5)=56x^6+12x^5[\/latex].<\/p>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<center><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script><\/center>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eafhcbeh-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-eafhcbeh-ruACLHzWT3g\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6246502&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-eafhcbeh-ruACLHzWT3g&vembed=0&video_id=ruACLHzWT3g&video_target=tpm-plugin-eafhcbeh-ruACLHzWT3g'><\/script><\/p>\r\nWatch the clip from 11:25 until 12:31. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules685to751_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739251962\">Find the derivative of [latex]g(x)=\\dfrac{1}{x^7}[\/latex] using the extended power rule.<\/p>\r\n[reveal-answer q=\"873564\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"873564\"]\r\n<p id=\"fs-id1169739252030\">Rewrite [latex]g(x)=\\frac{1}{x^7}=x^{-7}[\/latex]. Use the extended power rule with [latex]k=-7[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739251993\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739251993\"]\r\n<p id=\"fs-id1169739251993\">[latex]g^{\\prime}(x)=-7x^{-8}[\/latex].<\/p>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\" style=\"text-align: center;\"><iframe id=\"tpm-plugin-ebgggefg-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ebgggefg-ruACLHzWT3g\" class=\"p3sdk-target\" style=\"text-align: center;\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6246502&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ebgggefg-ruACLHzWT3g&vembed=0&video_id=ruACLHzWT3g&video_target=tpm-plugin-ebgggefg-ruACLHzWT3g'><\/script><\/p>\r\nWatch the clip from 18:45 until 19:08. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules1125to1148_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]h(x) = (x^3 + 2x)(4x^2 - 3)[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"633733\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"633733\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Let [latex]f(x) = x^3 + 2x[\/latex] and [latex]g(x) = 4x^2 - 3[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} f'(x) &amp;=&amp; 3x^2 + 2 \\\\ g'(x) &amp;=&amp; 8x \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Applying the product rule:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} h'(x) &amp;=&amp; f'(x)g(x) + g'(x)f(x) \\\\ &amp;=&amp; (3x^2 + 2)(4x^2 - 3) + (8x)(x^3 + 2x) \\\\ &amp;=&amp; (12x^4 - 9x^2 + 8x^2 - 6) + (8x^4 + 16x^2) \\\\ &amp;=&amp; 20x^4 + 15x^2 - 6 \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]k(x) = \\frac{x^2 + 3x}{2x - 1}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"762054\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"762054\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Let [latex]f(x) = x^2 + 3x[\/latex] and [latex]g(x) = 2x - 1[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} f'(x) &amp;=&amp; 2x + 3 \\\\ g'(x) &amp;=&amp; 2 \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Applying the quotient rule:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} k'(x) &amp;=&amp; \\dfrac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2} \\\\ &amp;=&amp; \\dfrac{(2x + 3)(2x - 1) - 2(x^2 + 3x)}{(2x - 1)^2} \\\\ &amp;=&amp; \\dfrac{4x^2 - 2x + 6x - 3 - 2x^2 - 6x}{(2x - 1)^2} \\\\ &amp;=&amp; \\dfrac{2x^2 - 2x - 3}{(2x - 1)^2} \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = 5x^{-3} - 2x^{-1}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[reveal-answer q=\"865573\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"865573\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Using the extended power rule and the sum rule:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} f'(x) &amp;=&amp; \\frac{d}{dx}(5x^{-3}) - \\frac{d}{dx}(2x^{-1}) \\\\ &amp;=&amp; 5(-3x^{-4}) - 2(-1x^{-2}) \\\\ &amp;=&amp; -15x^{-4} + 2x^{-2} \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n\r\n<\/section>\r\n<h2>Combining Differentiation Rules<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combining Multiple Rules:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Most real-world problems require applying several differentiation rules in sequence<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply rules in reverse order of function evaluation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Problem-Solving Strategy:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the structure of the function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine which rules apply and in what order<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the rules systematically<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The order of applying differentiation rules matters<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex functions often require a combination of product, quotient, and basic rules<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739325728\">For [latex]k(x)=f(x)g(x)h(x)[\/latex], express [latex]k^{\\prime}(x)[\/latex] in terms of [latex]f(x), \\, g(x), \\, h(x)[\/latex], and their derivatives.<\/p>\r\n[reveal-answer q=\"fs-id1169739270350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739270350\"]\r\n<p id=\"fs-id1169739270350\">We can think of the function [latex]k(x)[\/latex] as the product of the function [latex]f(x)g(x)[\/latex] and the function [latex]h(x)[\/latex]. That is, [latex]k(x)=(f(x)g(x))\\cdot h(x)[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1169739333852\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}k^{\\prime}(x) &amp; =\\frac{d}{dx}(f(x)g(x))\\cdot h(x)+\\frac{d}{dx}(h(x))\\cdot (f(x)g(x)) &amp; &amp; &amp; \\begin{array}{l}\\text{Apply the product rule to the product} \\\\ \\text{of} \\, f(x)g(x) \\, \\text{and} \\, h(x). \\end{array} \\\\ &amp; =(f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x))h(x)+h^{\\prime}(x)f(x)g(x) &amp; &amp; &amp; \\text{Apply the product rule to} \\, f(x)g(x). \\\\ &amp; =f^{\\prime}(x)g(x)h(x)+f(x)g^{\\prime}(x)h(x)+f(x)g(x)h^{\\prime}(x). &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<center><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script><\/center>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-edfagbde-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-edfagbde-ruACLHzWT3g\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6246502&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-edfagbde-ruACLHzWT3g&vembed=0&video_id=ruACLHzWT3g&video_target=tpm-plugin-edfagbde-ruACLHzWT3g'><\/script><\/p>\r\nWatch the clip from 20:25 until 22:39. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules1225to1359_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169736607620\">Find [latex]\\frac{d}{dx}(3f(x)-2g(x))[\/latex].<\/p>\r\n[reveal-answer q=\"288744\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"288744\"]\r\n<p id=\"fs-id1169736589229\">Apply the difference rule and the constant multiple rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736607671\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736607671\"]\r\n<p id=\"fs-id1169736607671\">[latex]3f^{\\prime}(x)-2g^{\\prime}(x)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739301467\">Find the value(s) of [latex]x[\/latex] for which the line tangent to the graph of [latex]f(x)=4x^2-3x+2[\/latex] is parallel to the line [latex]y=2x+3[\/latex].<\/p>\r\n[reveal-answer q=\"825443\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"825443\"]\r\n<p id=\"fs-id1169739298001\">Solve the equation [latex]f^{\\prime}(x)=2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739297983\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739297983\"]\r\n<p id=\"fs-id1169739297983\">[latex]\\frac{5}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the derivative of a function.<\/li>\n<li>Find instantaneous rates of change.<\/li>\n<li>Find an equation of the tangent line to the graph of a function at a point.<\/li>\n<\/ul>\n<\/section>\n<h2>Derivative Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Derivative Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Gives the derivative of a function at every point in its domain<\/li>\n<li class=\"whitespace-normal break-words\">Defined as: [latex]f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Differentiability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A function is differentiable at a point if its derivative exists at that point<\/li>\n<li class=\"whitespace-normal break-words\">A function is differentiable on an interval if it&#8217;s differentiable at every point in that interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Notation for Derivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(x)[\/latex], [latex]y'[\/latex], [latex]\\frac{dy}{dx}[\/latex], [latex]\\frac{d}{dx}(f(x))[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Leibniz Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Represents instantaneous rate of change<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169737819520\">Find the derivative of [latex]f(x)=x^2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737789025\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737789025\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737789025\">[latex]f^{\\prime}(x)=2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bcehahhc-w9m4h4i47-M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/w9m4h4i47-M?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bcehahhc-w9m4h4i47-M\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6246501&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bcehahhc-w9m4h4i47-M&#38;vembed=0&#38;video_id=w9m4h4i47-M&#38;video_target=tpm-plugin-bcehahhc-w9m4h4i47-M\"><\/script><\/p>\n<p style=\"text-align: left;\">Watch the clip from 5:49 until 6:48. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\n<p style=\"text-align: left;\">You can view the\u00a0<a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.2TheDerivativeAsAFunction349to408_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.2 The Derivative as a Function&#8221; here (opens in new window).<\/a> If you would like to watch the entire video, you can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/3.2+The+Derivative+as+a+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c3.2 The Derivative as a Function\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the derivative of [latex]f(x) = x^3 + 2x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q255451\">Show Answer<\/button><\/p>\n<div id=\"q255451\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} f'(x) &=& \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\\\ &=& \\lim_{h \\to 0} \\frac{((x+h)^3 + 2(x+h)) - (x^3 + 2x)}{h} \\\\ &=& \\lim_{h \\to 0} \\frac{(x^3 + 3x^2h + 3xh^2 + h^3 + 2x + 2h) - (x^3 + 2x)}{h} \\\\ &=& \\lim_{h \\to 0} \\frac{3x^2h + 3xh^2 + h^3 + 2h}{h} \\\\ &=& \\lim_{h \\to 0} (3x^2 + 3xh + h^2 + 2) \\\\ &=& 3x^2 + 2 \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]f'(x) = 3x^2 + 2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-badcaeeh-N2PpRnFqnqY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/N2PpRnFqnqY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-badcaeeh-N2PpRnFqnqY\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661480&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-badcaeeh-N2PpRnFqnqY&#38;vembed=0&#38;video_id=N2PpRnFqnqY&#38;video_target=tpm-plugin-badcaeeh-N2PpRnFqnqY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Derivative+as+a+concept+%7C+Derivatives+introduction+%7C+AP+Calculus+AB+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDerivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section aria-label=\"Watch It\">\n<h2>The Basic Rules<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Constant Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For any constant [latex]c[\/latex], [latex]\\frac{d}{dx}(c) = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Power Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]f(x) = x^n[\/latex] where [latex]n[\/latex] is a positive integer: [latex]\\frac{d}{dx}(x^n) = nx^{n-1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sum and Difference Rules:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Constant Multiple Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For any constant [latex]k[\/latex], [latex]\\frac{d}{dx}(kf(x)) = kf'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">These rules form the foundation for differentiating more complex functions.<\/li>\n<li class=\"whitespace-normal break-words\">The Power Rule applies to positive integer exponents and will be extended to other exponents later.<\/li>\n<li class=\"whitespace-normal break-words\">These rules allow us to differentiate polynomials and many other functions without using the limit definition every time.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736614166\">Find the derivative of [latex]g(x)=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738853102\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738853102\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738853102\">[latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738916812\">Find [latex]\\frac{d}{dx}(x^4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q41137798\">Hint<\/button><\/p>\n<div id=\"q41137798\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739270315\">Use [latex](x+h)^4=x^4+4x^3h+6x^2h^2+4xh^3+h^4[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739000891\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739000891\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739000891\">[latex]4x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739199747\">Find the derivative of [latex]f(x)=x^7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q25547709\">Hint<\/button><\/p>\n<div id=\"q25547709\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736613524\">Use the power rule with [latex]n=7[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738962015\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738962015\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738962015\">[latex]f^{\\prime}(x)=7x^6[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\" style=\"text-align: center;\"><iframe id=\"tpm-plugin-aeabafae-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aeabafae-ruACLHzWT3g\" class=\"p3sdk-target\" style=\"text-align: center;\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6246502&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aeabafae-ruACLHzWT3g&#38;vembed=0&#38;video_id=ruACLHzWT3g&#38;video_target=tpm-plugin-aeabafae-ruACLHzWT3g\"><\/script><\/p>\n<p>Watch the clip from 3:01 until 3:11. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules181to191_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>. You can also access the entire <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/3.3+Differentiation+Rules_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c3.3 Differentiation Rules\u201d here (opens in new window).<\/a><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739299469\">Find the derivative of [latex]f(x)=2x^3-6x^2+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736658726\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736658726\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658726\">[latex]f^{\\prime}(x)=6x^2-12x[\/latex].<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\" style=\"text-align: center;\"><iframe id=\"tpm-plugin-dfbdgbbc-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dfbdgbbc-ruACLHzWT3g\" class=\"p3sdk-target\" style=\"text-align: center;\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6246502&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dfbdgbbc-ruACLHzWT3g&#38;vembed=0&#38;video_id=ruACLHzWT3g&#38;video_target=tpm-plugin-dfbdgbbc-ruACLHzWT3g\"><\/script><\/p>\n<p>Watch the clip from 5:00 until 5:30. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules300to330_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736654291\">Find the equation of the line tangent to the graph of [latex]f(x)=3x^2-11[\/latex] at [latex]x=2[\/latex]. Use the point-slope form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739353706\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739353706\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739353706\">[latex]y=12x-23[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2 class=\"entry-title\">The Advanced Rules<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Product Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]j(x) = f(x)g(x)[\/latex]: [latex]j'(x) = f'(x)g(x) + g'(x)f(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Quotient Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]j(x) = \\frac{f(x)}{g(x)}[\/latex]: [latex]j'(x) = \\frac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Extended Power Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]k[\/latex] a negative integer: [latex]\\frac{d}{dx}(x^k) = kx^{k-1}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The product rule is not simply the product of the derivatives.<\/li>\n<li class=\"whitespace-normal break-words\">The quotient rule involves a specific arrangement of terms in the numerator.<\/li>\n<li class=\"whitespace-normal break-words\">The extended power rule allows differentiation of negative integer powers.<\/li>\n<li class=\"whitespace-normal break-words\">These rules expand our ability to differentiate more complex functions.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736654828\">Use the product rule to obtain the derivative of [latex]j(x)=2x^5(4x^2+x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q034256\">Hint<\/button><\/p>\n<div id=\"q034256\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736609959\">Set [latex]f(x)=2x^5[\/latex] and [latex]g(x)=4x^2+x[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736654876\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736654876\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736654876\">[latex]j^{\\prime}(x)=10x^4(4x^2+x)+(8x+1)(2x^5)=56x^6+12x^5[\/latex].<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/div>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-eafhcbeh-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-eafhcbeh-ruACLHzWT3g\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6246502&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-eafhcbeh-ruACLHzWT3g&#38;vembed=0&#38;video_id=ruACLHzWT3g&#38;video_target=tpm-plugin-eafhcbeh-ruACLHzWT3g\"><\/script><\/p>\n<p>Watch the clip from 11:25 until 12:31. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules685to751_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739251962\">Find the derivative of [latex]g(x)=\\dfrac{1}{x^7}[\/latex] using the extended power rule.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q873564\">Hint<\/button><\/p>\n<div id=\"q873564\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739252030\">Rewrite [latex]g(x)=\\frac{1}{x^7}=x^{-7}[\/latex]. Use the extended power rule with [latex]k=-7[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739251993\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739251993\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739251993\">[latex]g^{\\prime}(x)=-7x^{-8}[\/latex].<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\" style=\"text-align: center;\"><iframe id=\"tpm-plugin-ebgggefg-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ebgggefg-ruACLHzWT3g\" class=\"p3sdk-target\" style=\"text-align: center;\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6246502&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ebgggefg-ruACLHzWT3g&#38;vembed=0&#38;video_id=ruACLHzWT3g&#38;video_target=tpm-plugin-ebgggefg-ruACLHzWT3g\"><\/script><\/p>\n<p>Watch the clip from 18:45 until 19:08. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules1125to1148_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]h(x) = (x^3 + 2x)(4x^2 - 3)[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q633733\">Show Answer<\/button><\/p>\n<div id=\"q633733\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Let [latex]f(x) = x^3 + 2x[\/latex] and [latex]g(x) = 4x^2 - 3[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} f'(x) &=& 3x^2 + 2 \\\\ g'(x) &=& 8x \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Applying the product rule:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} h'(x) &=& f'(x)g(x) + g'(x)f(x) \\\\ &=& (3x^2 + 2)(4x^2 - 3) + (8x)(x^3 + 2x) \\\\ &=& (12x^4 - 9x^2 + 8x^2 - 6) + (8x^4 + 16x^2) \\\\ &=& 20x^4 + 15x^2 - 6 \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]k(x) = \\frac{x^2 + 3x}{2x - 1}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q762054\">Show Answer<\/button><\/p>\n<div id=\"q762054\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Let [latex]f(x) = x^2 + 3x[\/latex] and [latex]g(x) = 2x - 1[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} f'(x) &=& 2x + 3 \\\\ g'(x) &=& 2 \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Applying the quotient rule:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} k'(x) &=& \\dfrac{f'(x)g(x) - g'(x)f(x)}{[g(x)]^2} \\\\ &=& \\dfrac{(2x + 3)(2x - 1) - 2(x^2 + 3x)}{(2x - 1)^2} \\\\ &=& \\dfrac{4x^2 - 2x + 6x - 3 - 2x^2 - 6x}{(2x - 1)^2} \\\\ &=& \\dfrac{2x^2 - 2x - 3}{(2x - 1)^2} \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = 5x^{-3} - 2x^{-1}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q865573\">Show Answer<\/button><\/p>\n<div id=\"q865573\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\">Using the extended power rule and the sum rule:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} f'(x) &=& \\frac{d}{dx}(5x^{-3}) - \\frac{d}{dx}(2x^{-1}) \\\\ &=& 5(-3x^{-4}) - 2(-1x^{-2}) \\\\ &=& -15x^{-4} + 2x^{-2} \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<h2>Combining Differentiation Rules<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combining Multiple Rules:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Most real-world problems require applying several differentiation rules in sequence<\/li>\n<li class=\"whitespace-normal break-words\">Apply rules in reverse order of function evaluation<\/li>\n<\/ul>\n<\/li>\n<li>Problem-Solving Strategy:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the structure of the function<\/li>\n<li class=\"whitespace-normal break-words\">Determine which rules apply and in what order<\/li>\n<li class=\"whitespace-normal break-words\">Apply the rules systematically<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the result<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The order of applying differentiation rules matters<\/li>\n<li class=\"whitespace-normal break-words\">Complex functions often require a combination of product, quotient, and basic rules<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739325728\">For [latex]k(x)=f(x)g(x)h(x)[\/latex], express [latex]k^{\\prime}(x)[\/latex] in terms of [latex]f(x), \\, g(x), \\, h(x)[\/latex], and their derivatives.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739270350\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739270350\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739270350\">We can think of the function [latex]k(x)[\/latex] as the product of the function [latex]f(x)g(x)[\/latex] and the function [latex]h(x)[\/latex]. That is, [latex]k(x)=(f(x)g(x))\\cdot h(x)[\/latex]. Thus,<\/p>\n<div id=\"fs-id1169739333852\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}k^{\\prime}(x) & =\\frac{d}{dx}(f(x)g(x))\\cdot h(x)+\\frac{d}{dx}(h(x))\\cdot (f(x)g(x)) & & & \\begin{array}{l}\\text{Apply the product rule to the product} \\\\ \\text{of} \\, f(x)g(x) \\, \\text{and} \\, h(x). \\end{array} \\\\ & =(f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x))h(x)+h^{\\prime}(x)f(x)g(x) & & & \\text{Apply the product rule to} \\, f(x)g(x). \\\\ & =f^{\\prime}(x)g(x)h(x)+f(x)g^{\\prime}(x)h(x)+f(x)g(x)h^{\\prime}(x). & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/div>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-edfagbde-ruACLHzWT3g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-edfagbde-ruACLHzWT3g\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6246502&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-edfagbde-ruACLHzWT3g&#38;vembed=0&#38;video_id=ruACLHzWT3g&#38;video_target=tpm-plugin-edfagbde-ruACLHzWT3g\"><\/script><\/p>\n<p>Watch the clip from 20:25 until 22:39. If you prefer, you can also watch the video on its original page by clicking the Youtube logo in the lower left-hand corner of the video display.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules1225to1359_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736607620\">Find [latex]\\frac{d}{dx}(3f(x)-2g(x))[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q288744\">Hint<\/button><\/p>\n<div id=\"q288744\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736589229\">Apply the difference rule and the constant multiple rule.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736607671\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736607671\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736607671\">[latex]3f^{\\prime}(x)-2g^{\\prime}(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739301467\">Find the value(s) of [latex]x[\/latex] for which the line tangent to the graph of [latex]f(x)=4x^2-3x+2[\/latex] is parallel to the line [latex]y=2x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q825443\">Hint<\/button><\/p>\n<div id=\"q825443\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739298001\">Solve the equation [latex]f^{\\prime}(x)=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739297983\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739297983\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739297983\">[latex]\\frac{5}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n","protected":false},"author":67,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"3.2 The Derivative as a Function\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/w9m4h4i47-M\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/N2PpRnFqnqY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"3.3 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