{"id":548,"date":"2025-10-09T15:20:58","date_gmt":"2025-10-09T15:20:58","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=548"},"modified":"2025-11-20T15:07:51","modified_gmt":"2025-11-20T15:07:51","slug":"vector-valued-functions-and-motion-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-valued-functions-and-motion-background-youll-need-2\/","title":{"raw":"Vector-Valued Functions and Motion: Background You'll Need 2","rendered":"Vector-Valued Functions and Motion: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Basic Rules<\/h2>\r\n<h3>The Constant Rule<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The derivative of any constant is zero:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(c) = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Why? A constant function [latex]f(x) = c[\/latex] is a horizontal line with slope [latex]0[\/latex]. Since the derivative measures rate of change, and a constant never changes, its derivative is [latex]0[\/latex].<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">the constant rule<\/h3>\r\n<p id=\"fs-id1169738878658\">Let [latex]c[\/latex] be a constant.<\/p>\r\n<p id=\"fs-id1169738853363\" style=\"text-align: center;\">If [latex]f(x)=c[\/latex], then [latex]f^{\\prime}(c)=0[\/latex]<\/p>\r\n<p id=\"fs-id1169739024163\">Alternatively, we may express this rule as<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(c)=0[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169738824417\">Find the derivative of [latex]f(x)=8[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738865666\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738865666\"]\r\n<p id=\"fs-id1169738865666\">This is just a one-step application of the rule:<\/p>\r\n\r\n<div id=\"fs-id1169738875468\" class=\"equation unnumbered\">[latex]f^{\\prime}(8)=0[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3>The Power Rule<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">From Calculus 1, you know that differentiation follows predictable patterns for power functions. For example:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}\\left(x^2\\right) = 2x[\/latex] and [latex]\\dfrac{d}{dx}\\left(x^{\\frac{1}{2}}\\right) = \\dfrac{1}{2}x^{-\\frac{1}{2}}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Notice the pattern: the exponent becomes the coefficient, and the new exponent decreases by [latex]1[\/latex].<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">the power rule<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For any real number [latex]n[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(x^n) = nx^{n-1}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">This applies to all powers\u2014positive, negative, and fractional.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Examples:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{d}{dx}(x^3) = 3x^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{d}{dx}(x^{-2}) = -2x^{-3}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\dfrac{d}{dx}(x^{5\/3}) = \\dfrac{5}{3}x^{2\/3}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169736613648\">Find the derivative of the function [latex]f(x)=x^{10}[\/latex] by applying the power rule.<\/p>\r\n[reveal-answer q=\"fs-id1169736656146\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736656146\"]\r\n<p id=\"fs-id1169736656146\">Using the power rule with [latex]n=10[\/latex], we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739342007\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=10x^{10-1}=10x^9[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3>The Sum, Difference, and Constant Multiple Rules<\/h3>\r\n<p id=\"fs-id1169739270017\">We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">sum, difference, and constant multiple rules<\/h3>\r\n<p id=\"fs-id1169736611426\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions and [latex]k[\/latex] be a constant. Then each of the following equations holds.<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1169739039653\"><strong>Sum Rule:<\/strong>\u00a0The derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169739179597\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)+g(x))=\\frac{d}{dx}(f(x))+\\frac{d}{dx}(g(x))[\/latex];<\/div>\r\n<p id=\"fs-id1169739351562\">that is,<\/p>\r\n\r\n<div id=\"fs-id1169739008128\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)+g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169739000178\"><strong>Difference Rule:<\/strong>\u00a0The derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169736611311\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)-g(x))=\\frac{d}{dx}(f(x))-\\frac{d}{dx}(g(x))[\/latex];<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169739005930\">that is,<\/p>\r\n\r\n<div id=\"fs-id1169739208882\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)-g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)-g^{\\prime}(x)[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1169739195371\"><strong>Constant Multiple Rule:<\/strong>\u00a0The derivative of a constant [latex]k[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative:<\/p>\r\n\r\n<div id=\"fs-id1169739340266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}(f(x))[\/latex];<\/div>\r\n<p id=\"fs-id1169739179540\">that is,<\/p>\r\n\r\n<div id=\"fs-id1169739179543\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=kf(x), \\, j^{\\prime}(x)=kf^{\\prime}(x)[\/latex]<\/div>\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739305154\">Find the derivative of [latex]g(x)=3x^2[\/latex] and compare it to the derivative of [latex]f(x)=x^2[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739286421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739286421\"]\r\n<p id=\"fs-id1169739286421\">We use the power rule directly:<\/p>\r\n\r\n<div id=\"fs-id1169739286424\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\dfrac{d}{dx}(3x^2)=3\\dfrac{d}{dx}(x^2)=3(2x)=6x[\/latex].<\/div>\r\n<p id=\"fs-id1169736660712\">Since [latex]f(x)=x^2[\/latex] has derivative [latex]f^{\\prime}(x)=2x[\/latex], we see that the derivative of [latex]g(x)[\/latex] is [latex]3[\/latex] times the derivative of [latex]f(x)[\/latex]. This relationship is illustrated in the graphs below.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"859\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205331\/CNX_Calc_Figure_03_03_001.jpg\" alt=\"Two graphs are shown. The first graph shows g(x) = 3x2 and f(x) = x squared. The second graph shows g\u2019(x) = 6x and f\u2019(x) = 2x. In the first graph, g(x) increases three times more quickly than f(x). In the second graph, g\u2019(x) increases three times more quickly than f\u2019(x).\" width=\"859\" height=\"309\" \/> Figure 1. The derivative of [latex]g(x)[\/latex] is 3 times the derivative of [latex]f(x)[\/latex].[\/caption][\/hidden-answer]<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739300397\">Find the derivative of [latex]f(x)=2x^5+7[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739064774\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739064774\"]\r\n<p id=\"fs-id1169739064774\">We begin by applying the rule for differentiating the sum of two functions, followed by the rules for differentiating constant multiples of functions and the rule for differentiating powers. To better understand the sequence in which the differentiation rules are applied, we use Leibniz notation throughout the solution:<\/p>\r\n\r\n<div id=\"fs-id1169739273990\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}f^{\\prime}(x) &amp; =\\frac{d}{dx}(2x^5+7) &amp; &amp; &amp; \\\\ &amp; =\\frac{d}{dx}(2x^5)+\\frac{d}{dx}(7) &amp; &amp; &amp; \\text{Apply the sum rule.} \\\\ &amp; =2\\frac{d}{dx}(x^5)+\\frac{d}{dx}(7) &amp; &amp; &amp; \\text{Apply the constant multiple rule.} \\\\ &amp; =2(5x^4)+0 &amp; &amp; &amp; \\text{Apply the power rule and the constant rule.} \\\\ &amp; =10x^4. &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1169739242299\">Find the equation of the line tangent to the graph of [latex]f(x)=x^2-4x+6[\/latex] at [latex]x=1[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736663036\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736663036\"]\r\n<p id=\"fs-id1169736663036\">To find the equation of the tangent line, we need a point and a slope. To find the point, compute<\/p>\r\n\r\n<div id=\"fs-id1169736663039\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(1)=1^2-4(1)+6=3[\/latex].<\/div>\r\n<p id=\"fs-id1169736587931\">This gives us the point [latex](1,3)[\/latex]. Since the slope of the tangent line at [latex]1[\/latex] is [latex]f^{\\prime}(1)[\/latex], we must first find [latex]f^{\\prime}(x)[\/latex]. Using the definition of a derivative, we have<\/p>\r\n\r\n<div id=\"fs-id1169739297908\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=2x-4[\/latex]<\/div>\r\n<p id=\"fs-id1169739273132\">so the slope of the tangent line is [latex]f^{\\prime}(1)=-2[\/latex]. Using the point-slope formula, we see that the equation of the tangent line is<\/p>\r\n\r\n<div id=\"fs-id1169739273158\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y-3=-2(x-1)[\/latex].<\/div>\r\n<p id=\"fs-id1169739269639\">Putting the equation of the line in slope-intercept form, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739269643\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=-2x+5[\/latex].<\/div>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=338&amp;end=444&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>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.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules338to444_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 tryIt\">[ohm_question hide_question_numbers=1]33696[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use basic rules to differentiate functions that involve adding, subtracting, scaling, and raising to powers<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>The Basic Rules<\/h2>\n<h3>The Constant Rule<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The derivative of any constant is zero:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(c) = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Why? A constant function [latex]f(x) = c[\/latex] is a horizontal line with slope [latex]0[\/latex]. Since the derivative measures rate of change, and a constant never changes, its derivative is [latex]0[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">the constant rule<\/h3>\n<p id=\"fs-id1169738878658\">Let [latex]c[\/latex] be a constant.<\/p>\n<p id=\"fs-id1169738853363\" style=\"text-align: center;\">If [latex]f(x)=c[\/latex], then [latex]f^{\\prime}(c)=0[\/latex]<\/p>\n<p id=\"fs-id1169739024163\">Alternatively, we may express this rule as<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(c)=0[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738824417\">Find the derivative of [latex]f(x)=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738865666\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738865666\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738865666\">This is just a one-step application of the rule:<\/p>\n<div id=\"fs-id1169738875468\" class=\"equation unnumbered\">[latex]f^{\\prime}(8)=0[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>The Power Rule<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">From Calculus 1, you know that differentiation follows predictable patterns for power functions. For example:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}\\left(x^2\\right) = 2x[\/latex] and [latex]\\dfrac{d}{dx}\\left(x^{\\frac{1}{2}}\\right) = \\dfrac{1}{2}x^{-\\frac{1}{2}}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Notice the pattern: the exponent becomes the coefficient, and the new exponent decreases by [latex]1[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">the power rule<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For any real number [latex]n[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(x^n) = nx^{n-1}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">This applies to all powers\u2014positive, negative, and fractional.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Examples:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{d}{dx}(x^3) = 3x^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{d}{dx}(x^{-2}) = -2x^{-3}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\dfrac{d}{dx}(x^{5\/3}) = \\dfrac{5}{3}x^{2\/3}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169736613648\">Find the derivative of the function [latex]f(x)=x^{10}[\/latex] by applying the power rule.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736656146\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736656146\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736656146\">Using the power rule with [latex]n=10[\/latex], we obtain<\/p>\n<div id=\"fs-id1169739342007\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=10x^{10-1}=10x^9[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>The Sum, Difference, and Constant Multiple Rules<\/h3>\n<p id=\"fs-id1169739270017\">We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">sum, difference, and constant multiple rules<\/h3>\n<p id=\"fs-id1169736611426\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be differentiable functions and [latex]k[\/latex] be a constant. Then each of the following equations holds.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739039653\"><strong>Sum Rule:<\/strong>\u00a0The derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\n<div id=\"fs-id1169739179597\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)+g(x))=\\frac{d}{dx}(f(x))+\\frac{d}{dx}(g(x))[\/latex];<\/div>\n<p id=\"fs-id1169739351562\">that is,<\/p>\n<div id=\"fs-id1169739008128\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)+g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1169739000178\"><strong>Difference Rule:<\/strong>\u00a0The derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex].<\/p>\n<div id=\"fs-id1169736611311\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(f(x)-g(x))=\\frac{d}{dx}(f(x))-\\frac{d}{dx}(g(x))[\/latex];<\/div>\n<div><\/div>\n<p id=\"fs-id1169739005930\">that is,<\/p>\n<div id=\"fs-id1169739208882\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=f(x)-g(x), \\, j^{\\prime}(x)=f^{\\prime}(x)-g^{\\prime}(x)[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1169739195371\"><strong>Constant Multiple Rule:<\/strong>\u00a0The derivative of a constant [latex]k[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative:<\/p>\n<div id=\"fs-id1169739340266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(kf(x))=k\\frac{d}{dx}(f(x))[\/latex];<\/div>\n<p id=\"fs-id1169739179540\">that is,<\/p>\n<div id=\"fs-id1169739179543\" class=\"equation unnumbered\" style=\"text-align: center;\">for [latex]j(x)=kf(x), \\, j^{\\prime}(x)=kf^{\\prime}(x)[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739305154\">Find the derivative of [latex]g(x)=3x^2[\/latex] and compare it to 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-id1169739286421\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739286421\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739286421\">We use the power rule directly:<\/p>\n<div id=\"fs-id1169739286424\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\dfrac{d}{dx}(3x^2)=3\\dfrac{d}{dx}(x^2)=3(2x)=6x[\/latex].<\/div>\n<p id=\"fs-id1169736660712\">Since [latex]f(x)=x^2[\/latex] has derivative [latex]f^{\\prime}(x)=2x[\/latex], we see that the derivative of [latex]g(x)[\/latex] is [latex]3[\/latex] times the derivative of [latex]f(x)[\/latex]. This relationship is illustrated in the graphs below.<\/p>\n<figure style=\"width: 859px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205331\/CNX_Calc_Figure_03_03_001.jpg\" alt=\"Two graphs are shown. The first graph shows g(x) = 3x2 and f(x) = x squared. The second graph shows g\u2019(x) = 6x and f\u2019(x) = 2x. In the first graph, g(x) increases three times more quickly than f(x). In the second graph, g\u2019(x) increases three times more quickly than f\u2019(x).\" width=\"859\" height=\"309\" \/><figcaption class=\"wp-caption-text\">Figure 1. The derivative of [latex]g(x)[\/latex] is 3 times the derivative of [latex]f(x)[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739300397\">Find the derivative of [latex]f(x)=2x^5+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739064774\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739064774\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739064774\">We begin by applying the rule for differentiating the sum of two functions, followed by the rules for differentiating constant multiples of functions and the rule for differentiating powers. To better understand the sequence in which the differentiation rules are applied, we use Leibniz notation throughout the solution:<\/p>\n<div id=\"fs-id1169739273990\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}f^{\\prime}(x) & =\\frac{d}{dx}(2x^5+7) & & & \\\\ & =\\frac{d}{dx}(2x^5)+\\frac{d}{dx}(7) & & & \\text{Apply the sum rule.} \\\\ & =2\\frac{d}{dx}(x^5)+\\frac{d}{dx}(7) & & & \\text{Apply the constant multiple rule.} \\\\ & =2(5x^4)+0 & & & \\text{Apply the power rule and the constant rule.} \\\\ & =10x^4. & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169739242299\">Find the equation of the line tangent to the graph of [latex]f(x)=x^2-4x+6[\/latex] at [latex]x=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736663036\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736663036\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736663036\">To find the equation of the tangent line, we need a point and a slope. To find the point, compute<\/p>\n<div id=\"fs-id1169736663039\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(1)=1^2-4(1)+6=3[\/latex].<\/div>\n<p id=\"fs-id1169736587931\">This gives us the point [latex](1,3)[\/latex]. Since the slope of the tangent line at [latex]1[\/latex] is [latex]f^{\\prime}(1)[\/latex], we must first find [latex]f^{\\prime}(x)[\/latex]. Using the definition of a derivative, we have<\/p>\n<div id=\"fs-id1169739297908\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=2x-4[\/latex]<\/div>\n<p id=\"fs-id1169739273132\">so the slope of the tangent line is [latex]f^{\\prime}(1)=-2[\/latex]. Using the point-slope formula, we see that the equation of the tangent line is<\/p>\n<div id=\"fs-id1169739273158\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y-3=-2(x-1)[\/latex].<\/div>\n<p id=\"fs-id1169739269639\">Putting the equation of the line in slope-intercept form, we obtain<\/p>\n<div id=\"fs-id1169739269643\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=-2x+5[\/latex].<\/div>\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=338&amp;end=444&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\/3.3DifferentiationRules338to444_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 tryIt\"><iframe loading=\"lazy\" id=\"ohm33696\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33696&theme=lumen&iframe_resize_id=ohm33696&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-548","chapter","type-chapter","status-publish","hentry"],"part":424,"module-header":"background_you_need","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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/548\/revisions"}],"predecessor-version":[{"id":2619,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/548\/revisions\/2619"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/424"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/548\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=548"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=548"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=548"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}