{"id":2872,"date":"2024-06-10T18:42:28","date_gmt":"2024-06-10T18:42:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2872"},"modified":"2024-08-05T01:52:16","modified_gmt":"2024-08-05T01:52:16","slug":"differentiation-rules-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/differentiation-rules-fresh-take\/","title":{"raw":"Differentiation Rules: Fresh Take","rendered":"Differentiation Rules: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers<\/li>\r\n\t<li>Apply specific rules to find derivatives of functions multiplied or divided by each other<\/li>\r\n\t<li>Use a combination of rules to calculate derivatives for polynomial and rational functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Basic Rules<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\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\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\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\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\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<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<p>[reveal-answer q=\"fs-id1169738853102\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738853102\"]<\/p>\r\n<p id=\"fs-id1169738853102\">[latex]0[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738916812\">Find [latex]\\frac{d}{dx}(x^4)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"41137798\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"41137798\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739000891\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739000891\"]<\/p>\r\n<p id=\"fs-id1169739000891\">[latex]4x^3[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169739199747\">Find the derivative of [latex]f(x)=x^7[\/latex].<\/p>\r\n<p>[reveal-answer q=\"25547709\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"25547709\"]<\/p>\r\n<p id=\"fs-id1169736613524\">Use the power rule with [latex]n=7[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738962015\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738962015\"]<\/p>\r\n<p id=\"fs-id1169738962015\">[latex]f^{\\prime}(x)=7x^6[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=181&amp;end=191&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\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 \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<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<p>[reveal-answer q=\"fs-id1169736658726\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736658726\"]<\/p>\r\n<p id=\"fs-id1169736658726\">[latex]f^{\\prime}(x)=6x^2-12x[\/latex].<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=300&amp;end=330&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\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 \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<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<p>[reveal-answer q=\"fs-id1169739353706\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739353706\"]<\/p>\r\n<p id=\"fs-id1169739353706\">[latex]y=12x-23[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2 class=\"entry-title\">The Advanced Rules<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\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\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\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\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<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<p>[reveal-answer q=\"034256\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"034256\"]<\/p>\r\n<p id=\"fs-id1169736609959\">Set [latex]f(x)=2x^5[\/latex] and [latex]g(x)=4x^2+x[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736654876\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736654876\"]<\/p>\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\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=685&amp;end=751&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\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 \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<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<p>[reveal-answer q=\"873564\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"873564\"]<\/p>\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<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739251993\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739251993\"]<\/p>\r\n<p id=\"fs-id1169739251993\">[latex]g^{\\prime}(x)=-7x^{-8}[\/latex].<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1125&amp;end=1148&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\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 \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<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\"><br \/>\r\n[reveal-answer q=\"633733\"]Show Answer[\/reveal-answer]<br \/>\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<\/section>\r\n<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\"><br \/>\r\n[reveal-answer q=\"762054\"]Show Answer[\/reveal-answer]<br \/>\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<\/section>\r\n<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\"><br \/>\r\n[reveal-answer q=\"865573\"]Show Answer[\/reveal-answer]<br \/>\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<\/section>\r\n<h2>Combining Differentiation Rules<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\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\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\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<p>[reveal-answer q=\"fs-id1169739270350\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739270350\"]<\/p>\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<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<p>&nbsp;<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1225&amp;end=1359&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\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 \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169736607620\">Find [latex]\\frac{d}{dx}(3f(x)-2g(x))[\/latex].<\/p>\r\n<p>[reveal-answer q=\"288744\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"288744\"]<\/p>\r\n<p id=\"fs-id1169736589229\">Apply the difference rule and the constant multiple rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169736607671\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736607671\"]<\/p>\r\n<p id=\"fs-id1169736607671\">[latex]3f^{\\prime}(x)-2g^{\\prime}(x)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<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<p>[reveal-answer q=\"825443\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"825443\"]<\/p>\r\n<p id=\"fs-id1169739298001\">Solve the equation [latex]f^{\\prime}(x)=2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169739297983\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169739297983\"]<\/p>\r\n<p id=\"fs-id1169739297983\">[latex]\\frac{5}{8}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers<\/li>\n<li>Apply specific rules to find derivatives of functions multiplied or divided by each other<\/li>\n<li>Use a combination of rules to calculate derivatives for polynomial and rational functions<\/li>\n<\/ul>\n<\/section>\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<p><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<p><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<p><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<p><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<p><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<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=181&amp;end=191&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/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>.<\/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<p><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><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=300&amp;end=330&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/button><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/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<p><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<p><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<p><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;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=685&amp;end=751&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/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<p><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<p><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><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1125&amp;end=1148&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/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<div class=\"wp-nocaption \"><\/div>\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<div class=\"wp-nocaption \"><\/div>\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<div class=\"wp-nocaption \"><\/div>\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<p><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;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1225&amp;end=1359&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/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<p><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<p><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<p><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<p><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","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2872"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2872\/revisions"}],"predecessor-version":[{"id":3734,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2872\/revisions\/3734"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2872\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2872"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2872"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2872"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2872"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}