{"id":239,"date":"2023-09-20T22:48:30","date_gmt":"2023-09-20T22:48:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-basic-rules\/"},"modified":"2024-08-05T01:51:03","modified_gmt":"2024-08-05T01:51:03","slug":"differentiation-rules-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/differentiation-rules-learn-it-1\/","title":{"raw":"Differentiation Rules: Learn It 1","rendered":"Differentiation Rules: Learn It 1"},"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<p>Finding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process. In this section, we develop rules for finding derivatives that allow us to bypass this process. We begin with the basics.<br \/>\r\n<br \/>\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.<\/p>\r\n<h3>The Constant Rule<\/h3>\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<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<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.\u00a0<\/p>\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<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738824417\">Find the derivative of [latex]f(x)=8[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738865666\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738865666\"]<\/p>\r\n<p id=\"fs-id1169738865666\">This is just a one-step application of the rule:<\/p>\r\n<div id=\"fs-id1169738875468\" class=\"equation unnumbered\">[latex]f^{\\prime}(8)=0[\/latex].<\/div>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>The Power Rule<\/h3>\r\n<p id=\"fs-id1169739006200\">We have previously shown that,<\/p>\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<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]. <br \/>\r\n<br \/>\r\nWe 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. <br \/>\r\n<br \/>\r\nBefore 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<section class=\"textbox example\">\r\n<p id=\"fs-id1169739020915\">Find [latex]\\frac{d}{dx}(x^3)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738891154\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738891154\"]<\/p>\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<div class=\"equation unnumbered\" style=\"text-align: left;\">[\/hidden-answer]<\/div>\r\n<\/section>\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 straightforward. 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, and the new exponent decreases by [latex]1[\/latex], resulting in [latex]\\frac{d}{dx}(x^3)=3x^2[\/latex]<br \/>\r\n<br \/>\r\nThe following theorem states that this <strong>power rule<\/strong> holds for all positive integer powers of [latex]x[\/latex]. We will eventually extend this result to negative and rational powers of [latex]x[\/latex].\u00a0<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>Be aware that this rule does not apply to functions where a constant is raised to a variable power, such as [latex]f(x)=3^x[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">the power rule<\/h3>\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<div id=\"fs-id1169739269835\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=nx^{n-1}[\/latex]<\/div>\r\n<p>&nbsp;<\/p>\r\n<p id=\"fs-id1169739031758\">Alternatively, we may express this rule as<\/p>\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<\/section>\r\n<section class=\"textbox connectIt\">\r\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\r\n<hr \/>\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<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<p>&nbsp;<\/p>\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<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<p>&nbsp;<\/p>\r\n<p id=\"fs-id1169739030934\">Next, divide both sides by [latex]h[\/latex]:<\/p>\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<p>&nbsp;<\/p>\r\n<p id=\"fs-id1169738960216\">Thus,<\/p>\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<p>&nbsp;<\/p>\r\n<p id=\"fs-id1169739302943\">Finally,<\/p>\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<p>[latex]_\\blacksquare[\/latex]<\/p>\r\n<\/section>\r\n<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<p>[reveal-answer q=\"fs-id1169736656146\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169736656146\"]<\/p>\r\n<p id=\"fs-id1169736656146\">Using the power rule with [latex]n=10[\/latex], we obtain<\/p>\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<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<p>Finding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process. In this section, we develop rules for finding derivatives that allow us to bypass this process. We begin with the basics.<\/p>\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<h3>The Constant Rule<\/h3>\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 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.\u00a0<\/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<p><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 id=\"fs-id1169739006200\">We have previously 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 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]. <\/p>\n<p>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. <\/p>\n<p>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<section class=\"textbox example\">\n<p id=\"fs-id1169739020915\">Find [latex]\\frac{d}{dx}(x^3)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738891154\">Show Solution<\/button><\/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<div class=\"equation unnumbered\" style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/section>\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 straightforward. 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, and the new exponent decreases by [latex]1[\/latex], resulting in [latex]\\frac{d}{dx}(x^3)=3x^2[\/latex]<\/p>\n<p>The following theorem states that this <strong>power rule<\/strong> holds for all positive integer powers of [latex]x[\/latex]. We will eventually extend this result to negative and rational powers of [latex]x[\/latex].\u00a0<\/p>\n<section class=\"textbox proTip\">\n<p>Be aware that this rule does not apply to functions where a constant is raised to a variable power, such as [latex]f(x)=3^x[\/latex].<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">the power rule<\/h3>\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<\/section>\n<section class=\"textbox connectIt\">\n<p style=\"text-align: center;\"><strong>Proof<\/strong><\/p>\n<hr \/>\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 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]<br \/>\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<p><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","protected":false},"author":6,"menu_order":16,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus 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