{"id":335,"date":"2021-02-04T01:13:55","date_gmt":"2021-02-04T01:13:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=335"},"modified":"2022-03-16T05:29:18","modified_gmt":"2022-03-16T05:29:18","slug":"the-basic-rules","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-basic-rules\/","title":{"raw":"The Basic Rules","rendered":"The Basic Rules"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>State the constant, constant multiple, and power rules<\/li>\r\n \t<li>Apply the sum and difference rules to combine derivatives<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169738828810\" class=\"bc-section section\">\r\n<div id=\"fs-id1169739030384\" class=\"bc-section section\">\r\n\r\nThe functions [latex]f(x)=c[\/latex] and [latex]g(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.\r\n<h2>The Constant Rule<\/h2>\r\n<p id=\"fs-id1169738835554\">We first apply the limit definition of the derivative to find the derivative of the constant function, [latex]f(x)=c[\/latex]. For this function, both [latex]f(x)=c[\/latex] and [latex]f(x+h)=c[\/latex], so we obtain the following result:<\/p>\r\n\r\n<div id=\"fs-id1169738850732\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) &amp; =\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\dfrac{c-c}{h} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\dfrac{0}{h} \\\\ &amp; =\\underset{h\\to 0}{\\lim}0=0 \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738850205\">The rule for differentiating constant functions is called the <strong>constant rule<\/strong>. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is 0. We restate this rule in the following theorem.<\/p>\r\n\r\n<div id=\"fs-id1169738955425\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Constant Rule<\/h3>\r\n\r\n<hr \/>\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&nbsp;\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&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739274547\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Constant Rule<\/h3>\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169738954922\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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\">0<\/p>\r\n\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169738907507\">Use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Power Rule<\/h2>\r\n<p id=\"fs-id1169739006200\">We have shown that<\/p>\r\n\r\n<div id=\"fs-id1169739234394\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}\\left(x^2\\right)=2x[\/latex]\u00a0 \u00a0and\u00a0 \u00a0[latex]\\dfrac{d}{dx}\\left(x^{\\frac{1}{2}}\\right)=\\dfrac{1}{2}x^{\u2212\\frac{1}{2}}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738969429\">At this point, you might see a pattern beginning to develop for derivatives of the form [latex]\\frac{d}{dx}(x^n)[\/latex]. We continue our examination of derivative formulas by differentiating power functions of the form [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, [latex]\\frac{d}{dx}(x^3)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169738993994\" class=\"textbook exercises\">\r\n<h3>Example: Differentiating [latex]x^3[\/latex]<\/h3>\r\n<p id=\"fs-id1169739020915\">Find [latex]\\frac{d}{dx}(x^3)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738891154\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738891154\"]\r\n<div id=\"fs-id1169739014416\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\frac{d}{dx}(x^3) &amp; =\\underset{h\\to 0}{\\lim}\\frac{(x+h)^3-x^3}{h} &amp; &amp; &amp; \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} &amp; &amp; &amp; \\begin{array}{l}\\text{Notice that the first term in the expansion of} \\\\ (x+h)^3 \\, \\text{is} \\, x^3 \\, \\text{and the second term is} \\, 3x^2h. \\, \\text{All} \\\\ \\text{other terms contain powers of} \\, h \\, \\text{that are two or} \\\\ \\text{greater.} \\end{array} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{3x^2h+3xh^2+h^3}{h} &amp; &amp; &amp; \\begin{array}{l}\\text{In this step the} \\, x^3 \\, \\text{terms have been cancelled,} \\\\ \\text{leaving only terms containing} \\, h. \\end{array} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{h(3x^2+3xh+h^2)}{h} &amp; &amp; &amp; \\text{Factor out the common factor of} \\, h. \\\\ &amp; =\\underset{h\\to 0}{\\lim}(3x^2+3xh+h^2) &amp; &amp; &amp; \\begin{array}{l}\\text{After cancelling the common factor of} \\, h, \\, \\text{the} \\\\ \\text{only term not containing} \\, h \\, \\text{is} \\, 3x^2. \\end{array} \\\\ &amp; =3x^2 &amp; &amp; &amp; \\text{Let} \\, h \\, \\text{go to 0.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739302356\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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] and follow the procedure outlined in the preceding example.<\/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<\/div>\r\n<p id=\"fs-id1169736619689\">As we shall see, the procedure for finding the derivative of the general form [latex]f(x)=x^n[\/latex] is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate [latex]f(x)=x^3[\/latex], the power on [latex]x[\/latex] becomes the coefficient of [latex]x^2[\/latex] in the derivative and the power on [latex]x[\/latex] in the derivative decreases by 1. The following theorem states that this\u00a0<strong>power rule<\/strong> holds for all positive integer powers of [latex]x[\/latex]. We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of [latex]x[\/latex] and then to arbitrary powers of [latex]x[\/latex]. Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as [latex]f(x)=3^x[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169736615212\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">The Power Rule<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169738850005\">Let [latex]n[\/latex] be a positive integer. If [latex]f(x)=x^n[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1169739269835\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=nx^{n-1}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739031758\">Alternatively, we may express this rule as<\/p>\r\n\r\n<div id=\"fs-id1169739225629\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738999189\" class=\"bc-section section\">\r\n<h3>Proof<\/h3>\r\n<p id=\"fs-id1169738858013\">For [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer, we have<\/p>\r\n\r\n<div id=\"fs-id1169739010866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\frac{(x+h)^n-x^n}{h}[\/latex].<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div id=\"fs-id1169738108118\" class=\"equation unnumbered\">Since [latex](x+h)^n=x^n+nx^{n-1}h+\\binom{n}{2}x^{n-2}h^2+\\binom{n}{3}x^{n-3}h^3+\\cdots+nxh^{n-1}+h^n[\/latex],<\/div>\r\n<p id=\"fs-id1169738994258\">we see that<\/p>\r\n\r\n<div id=\"fs-id1169736614176\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex](x+h)^n-x^n=nx^{n-1}h+\\binom{n}{2}x^{n-2}h^2+\\binom{n}{3}x^{n-3}h^3+\\cdots+nxh^{n-1}+h^n[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739030934\">Next, divide both sides by [latex]h[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1169739190480\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\large \\frac{(x+h)^n-x^n}{h}=\\frac{nx^{n-1}h+\\binom{n}{2}x^{n-2}h^2+\\binom{n}{3}x^{n-3}h^3+\\cdots+nxh^{n-1}+h^n}{h}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738960216\">Thus,<\/p>\r\n\r\n<div id=\"fs-id1169739014535\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{(x+h)^n-x^n}{h}=nx^{n-1}+\\binom{n}{2}x^{n-2}h+\\binom{n}{3}x^{n-3}h^2+\\cdots+nxh^{n-2}+h^{n-1}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739302943\">Finally,<\/p>\r\n\r\n<div id=\"fs-id1169739302946\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) &amp; =\\underset{h\\to 0}{\\lim}(nx^{n-1}+\\binom{n}{2}x^{n-2}h+\\binom{n}{3}x^{n-3}h^2+\\cdots+nxh^{n-1}+h^n) \\\\ &amp; =nx^{n-1} \\end{array}[\/latex]<\/div>\r\n[latex]_\\blacksquare[\/latex]\r\n<div id=\"fs-id1169739190555\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Power Rule<\/h3>\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<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169736589163\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\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\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]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\/3.3DifferentiationRules181to191_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<h2>The Sum, Difference, and Constant Multiple Rules<\/h2>\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. These rules are summarized in the following theorem.<\/p>\r\n\r\n<div id=\"fs-id1169739305071\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Sum, Difference, and Constant Multiple Rules<\/h3>\r\n\r\n<hr \/>\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&nbsp;\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&nbsp;\r\n<div><\/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&nbsp;\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&nbsp;\r\n<div><\/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&nbsp;\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&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736619710\" class=\"bc-section section\">\r\n<h3>Proof<\/h3>\r\n<p id=\"fs-id1169736594897\">We provide only the proof of the sum rule here. The rest follow in a similar manner.<\/p>\r\n<p id=\"fs-id1169736659244\">For differentiable functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex], we set [latex]j(x)=f(x)+g(x)[\/latex]. Using the limit definition of the derivative we have<\/p>\r\n\r\n<div id=\"fs-id1169739269666\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{j(x+h)-j(x)}{h}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739343658\">By substituting [latex]j(x+h)=f(x+h)+g(x+h)[\/latex] and [latex]j(x)=f(x)+g(x)[\/latex], we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739274306\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{(f(x+h)+g(x+h))-(f(x)+g(x))}{h}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736613826\">Rearranging and regrouping the terms, we have<\/p>\r\n\r\n<div id=\"fs-id1169739303318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}(\\frac{f(x+h)-f(x)}{h}+\\frac{g(x+h)-g(x)}{h})[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739274627\">We now apply the sum law for limits and the definition of the derivative to obtain<\/p>\r\n\r\n<div id=\"fs-id1169739274631\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}(\\frac{f(x+h)-f(x)}{h})+\\underset{h\\to 0}{\\lim}(\\frac{g(x+h)-g(x)}{h})=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/div>\r\n[latex]_\\blacksquare[\/latex]\r\n<div id=\"fs-id1169739269764\" class=\"textbook exercises\">\r\n<h3>Example: Applying the Constant Multiple Rule<\/h3>\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&nbsp;\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 3 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]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739300387\" class=\"textbook exercises\">\r\n<h3>Example: Applying Basic Derivative Rules<\/h3>\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<\/div>\r\n<div id=\"fs-id1169739304168\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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=\"990033\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"990033\"]\r\n<p id=\"fs-id1169739301883\">Use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\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\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\r\n<center><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><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\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.\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>.[\/hidden-answer]\r\n<div id=\"fs-id1169739301889\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Equation of a Tangent Line<\/h3>\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&nbsp;\r\n<p id=\"fs-id1169736587931\">This gives us the point [latex](1,3)[\/latex]. Since the slope of the tangent line at 1 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&nbsp;\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&nbsp;\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\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Finding the Equation of a Tangent Line.\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\"><\/iframe><\/center>[reveal-answer q=\"266835\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266835\"]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>.[\/hidden-answer]\r\n<div id=\"fs-id1169736654283\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\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\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1169739353731\">Use the preceding example as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]33696[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>State the constant, constant multiple, and power rules<\/li>\n<li>Apply the sum and difference rules to combine derivatives<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169738828810\" class=\"bc-section section\">\n<div id=\"fs-id1169739030384\" class=\"bc-section section\">\n<p>The functions [latex]f(x)=c[\/latex] and [latex]g(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer are the building blocks from which all polynomials and rational functions are constructed. To find derivatives of polynomials and rational functions efficiently without resorting to the limit definition of the derivative, we must first develop formulas for differentiating these basic functions.<\/p>\n<h2>The Constant Rule<\/h2>\n<p id=\"fs-id1169738835554\">We first apply the limit definition of the derivative to find the derivative of the constant function, [latex]f(x)=c[\/latex]. For this function, both [latex]f(x)=c[\/latex] and [latex]f(x+h)=c[\/latex], so we obtain the following result:<\/p>\n<div id=\"fs-id1169738850732\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) & =\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h} \\\\ & =\\underset{h\\to 0}{\\lim}\\dfrac{c-c}{h} \\\\ & =\\underset{h\\to 0}{\\lim}\\dfrac{0}{h} \\\\ & =\\underset{h\\to 0}{\\lim}0=0 \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738850205\">The rule for differentiating constant functions is called the <strong>constant rule<\/strong>. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is 0. We restate this rule in the following theorem.<\/p>\n<div id=\"fs-id1169738955425\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Constant Rule<\/h3>\n<hr \/>\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>&nbsp;<\/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<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169739274547\" class=\"textbook exercises\">\n<h3>Example: Applying the Constant Rule<\/h3>\n<p id=\"fs-id1169738824417\">Find the derivative of [latex]f(x)=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738865666\">Show Solution<\/span><\/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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738954922\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736614166\">Find the derivative of [latex]g(x)=-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738853102\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738853102\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738853102\">0<\/p>\n<h4>Hint<\/h4>\n<p id=\"fs-id1169738907507\">Use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Power Rule<\/h2>\n<p id=\"fs-id1169739006200\">We have shown that<\/p>\n<div id=\"fs-id1169739234394\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}\\left(x^2\\right)=2x[\/latex]\u00a0 \u00a0and\u00a0 \u00a0[latex]\\dfrac{d}{dx}\\left(x^{\\frac{1}{2}}\\right)=\\dfrac{1}{2}x^{\u2212\\frac{1}{2}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738969429\">At this point, you might see a pattern beginning to develop for derivatives of the form [latex]\\frac{d}{dx}(x^n)[\/latex]. We continue our examination of derivative formulas by differentiating power functions of the form [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer. We develop formulas for derivatives of this type of function in stages, beginning with positive integer powers. Before stating and proving the general rule for derivatives of functions of this form, we take a look at a specific case, [latex]\\frac{d}{dx}(x^3)[\/latex].<\/p>\n<div id=\"fs-id1169738993994\" class=\"textbook exercises\">\n<h3>Example: Differentiating [latex]x^3[\/latex]<\/h3>\n<p id=\"fs-id1169739020915\">Find [latex]\\frac{d}{dx}(x^3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738891154\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738891154\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169739014416\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}\\frac{d}{dx}(x^3) & =\\underset{h\\to 0}{\\lim}\\frac{(x+h)^3-x^3}{h} & & & \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} & & & \\begin{array}{l}\\text{Notice that the first term in the expansion of} \\\\ (x+h)^3 \\, \\text{is} \\, x^3 \\, \\text{and the second term is} \\, 3x^2h. \\, \\text{All} \\\\ \\text{other terms contain powers of} \\, h \\, \\text{that are two or} \\\\ \\text{greater.} \\end{array} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{3x^2h+3xh^2+h^3}{h} & & & \\begin{array}{l}\\text{In this step the} \\, x^3 \\, \\text{terms have been cancelled,} \\\\ \\text{leaving only terms containing} \\, h. \\end{array} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{h(3x^2+3xh+h^2)}{h} & & & \\text{Factor out the common factor of} \\, h. \\\\ & =\\underset{h\\to 0}{\\lim}(3x^2+3xh+h^2) & & & \\begin{array}{l}\\text{After cancelling the common factor of} \\, h, \\, \\text{the} \\\\ \\text{only term not containing} \\, h \\, \\text{is} \\, 3x^2. \\end{array} \\\\ & =3x^2 & & & \\text{Let} \\, h \\, \\text{go to 0.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739302356\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169738916812\">Find [latex]\\frac{d}{dx}(x^4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q41137798\">Hint<\/span><\/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] and follow the procedure outlined in the preceding example.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739000891\">Show Solution<\/span><\/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<\/div>\n<p id=\"fs-id1169736619689\">As we shall see, the procedure for finding the derivative of the general form [latex]f(x)=x^n[\/latex] is very similar. Although it is often unwise to draw general conclusions from specific examples, we note that when we differentiate [latex]f(x)=x^3[\/latex], the power on [latex]x[\/latex] becomes the coefficient of [latex]x^2[\/latex] in the derivative and the power on [latex]x[\/latex] in the derivative decreases by 1. The following theorem states that this\u00a0<strong>power rule<\/strong> holds for all positive integer powers of [latex]x[\/latex]. We will eventually extend this result to negative integer powers. Later, we will see that this rule may also be extended first to rational powers of [latex]x[\/latex] and then to arbitrary powers of [latex]x[\/latex]. Be aware, however, that this rule does not apply to functions in which a constant is raised to a variable power, such as [latex]f(x)=3^x[\/latex].<\/p>\n<div id=\"fs-id1169736615212\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">The Power Rule<\/h3>\n<hr \/>\n<p id=\"fs-id1169738850005\">Let [latex]n[\/latex] be a positive integer. If [latex]f(x)=x^n[\/latex], then<\/p>\n<div id=\"fs-id1169739269835\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=nx^{n-1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739031758\">Alternatively, we may express this rule as<\/p>\n<div id=\"fs-id1169739225629\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169738999189\" class=\"bc-section section\">\n<h3>Proof<\/h3>\n<p id=\"fs-id1169738858013\">For [latex]f(x)=x^n[\/latex] where [latex]n[\/latex] is a positive integer, we have<\/p>\n<div id=\"fs-id1169739010866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\frac{(x+h)^n-x^n}{h}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div id=\"fs-id1169738108118\" class=\"equation unnumbered\">Since [latex](x+h)^n=x^n+nx^{n-1}h+\\binom{n}{2}x^{n-2}h^2+\\binom{n}{3}x^{n-3}h^3+\\cdots+nxh^{n-1}+h^n[\/latex],<\/div>\n<p id=\"fs-id1169738994258\">we see that<\/p>\n<div id=\"fs-id1169736614176\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex](x+h)^n-x^n=nx^{n-1}h+\\binom{n}{2}x^{n-2}h^2+\\binom{n}{3}x^{n-3}h^3+\\cdots+nxh^{n-1}+h^n[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739030934\">Next, divide both sides by [latex]h[\/latex]:<\/p>\n<div id=\"fs-id1169739190480\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\large \\frac{(x+h)^n-x^n}{h}=\\frac{nx^{n-1}h+\\binom{n}{2}x^{n-2}h^2+\\binom{n}{3}x^{n-3}h^3+\\cdots+nxh^{n-1}+h^n}{h}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738960216\">Thus,<\/p>\n<div id=\"fs-id1169739014535\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{(x+h)^n-x^n}{h}=nx^{n-1}+\\binom{n}{2}x^{n-2}h+\\binom{n}{3}x^{n-3}h^2+\\cdots+nxh^{n-2}+h^{n-1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739302943\">Finally,<\/p>\n<div id=\"fs-id1169739302946\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) & =\\underset{h\\to 0}{\\lim}(nx^{n-1}+\\binom{n}{2}x^{n-2}h+\\binom{n}{3}x^{n-3}h^2+\\cdots+nxh^{n-1}+h^n) \\\\ & =nx^{n-1} \\end{array}[\/latex]<\/div>\n<p>[latex]_\\blacksquare[\/latex]<\/p>\n<div id=\"fs-id1169739190555\" class=\"textbook exercises\">\n<h3>Example: Applying the Power Rule<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736656146\">Show Solution<\/span><\/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<\/div>\n<\/div>\n<div id=\"fs-id1169736589163\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739199747\">Find the derivative of [latex]f(x)=x^7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q25547709\">Hint<\/span><\/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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738962015\">Show Solution<\/span><\/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<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It.<\/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\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">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>.<\/div>\n<\/div>\n<h2>The Sum, Difference, and Constant Multiple Rules<\/h2>\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. These rules are summarized in the following theorem.<\/p>\n<div id=\"fs-id1169739305071\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Sum, Difference, and Constant Multiple Rules<\/h3>\n<hr \/>\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>&nbsp;<\/p>\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<p>&nbsp;<\/p>\n<div><\/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<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\n<div><\/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>&nbsp;<\/p>\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<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169736619710\" class=\"bc-section section\">\n<h3>Proof<\/h3>\n<p id=\"fs-id1169736594897\">We provide only the proof of the sum rule here. The rest follow in a similar manner.<\/p>\n<p id=\"fs-id1169736659244\">For differentiable functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex], we set [latex]j(x)=f(x)+g(x)[\/latex]. Using the limit definition of the derivative we have<\/p>\n<div id=\"fs-id1169739269666\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{j(x+h)-j(x)}{h}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739343658\">By substituting [latex]j(x+h)=f(x+h)+g(x+h)[\/latex] and [latex]j(x)=f(x)+g(x)[\/latex], we obtain<\/p>\n<div id=\"fs-id1169739274306\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{(f(x+h)+g(x+h))-(f(x)+g(x))}{h}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736613826\">Rearranging and regrouping the terms, we have<\/p>\n<div id=\"fs-id1169739303318\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}(\\frac{f(x+h)-f(x)}{h}+\\frac{g(x+h)-g(x)}{h})[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739274627\">We now apply the sum law for limits and the definition of the derivative to obtain<\/p>\n<div id=\"fs-id1169739274631\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]j^{\\prime}(x)=\\underset{h\\to 0}{\\lim}(\\frac{f(x+h)-f(x)}{h})+\\underset{h\\to 0}{\\lim}(\\frac{g(x+h)-g(x)}{h})=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/div>\n<p>[latex]_\\blacksquare[\/latex]<\/p>\n<div id=\"fs-id1169739269764\" class=\"textbook exercises\">\n<h3>Example: Applying the Constant Multiple Rule<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739286421\">Show Solution<\/span><\/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>&nbsp;<\/p>\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 3 times the derivative of [latex]f(x)[\/latex]. This relationship is illustrated in the graphs below.<\/p>\n<div style=\"width: 869px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The derivative of [latex]g(x)[\/latex] is 3 times the derivative of [latex]f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739300387\" class=\"textbook exercises\">\n<h3>Example: Applying Basic Derivative Rules<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739064774\">Show Solution<\/span><\/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<\/div>\n<div id=\"fs-id1169739304168\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990033\">Hint<\/span><\/p>\n<div id=\"q990033\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739301883\">Use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658726\">Show Solution<\/span><\/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<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It.<\/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=300&amp;end=330&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">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>.<\/div>\n<\/div>\n<div id=\"fs-id1169739301889\" class=\"textbook exercises\">\n<h3>Example: Finding the Equation of a Tangent Line<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736663036\">Show Solution<\/span><\/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>&nbsp;<\/p>\n<p id=\"fs-id1169736587931\">This gives us the point [latex](1,3)[\/latex]. Since the slope of the tangent line at 1 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>&nbsp;<\/p>\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>&nbsp;<\/p>\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<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Finding the Equation of a Tangent Line.<\/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\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266835\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266835\" class=\"hidden-answer\" style=\"display: none\">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>.<\/div>\n<\/div>\n<div id=\"fs-id1169736654283\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739353706\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739353706\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739353706\">[latex]y=12x-23[\/latex]<\/p>\n<h4>Hint<\/h4>\n<p id=\"fs-id1169739353731\">Use the preceding example as a guide.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm33696\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33696&theme=oea&iframe_resize_id=ohm33696&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-335\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.3 Differentiation Rules. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.3 Differentiation Rules\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-335","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/335","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":29,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/335\/revisions"}],"predecessor-version":[{"id":4806,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/335\/revisions\/4806"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/35"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/335\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=335"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=335"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=335"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}