{"id":3372,"date":"2024-06-20T15:42:18","date_gmt":"2024-06-20T15:42:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3372"},"modified":"2024-08-05T02:04:37","modified_gmt":"2024-08-05T02:04:37","slug":"derivatives-of-exponential-and-logarithmic-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-of-exponential-and-logarithmic-functions-fresh-take\/","title":{"raw":"Derivatives of Exponential and Logarithmic Functions: Fresh Take","rendered":"Derivatives of Exponential and Logarithmic Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Determine the derivatives of exponential and logarithmic functions<\/li>\r\n\t<li>Apply logarithmic differentiation to find derivatives<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivative of the Exponential Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">The Natural Exponential Function:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Defined as [latex]E(x) = e^x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]e \\approx 2.718281828...[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Property:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^x) = e^x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">General Exponential Function:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]B(x) = b^x[\/latex] where [latex]b &gt; 0[\/latex]: [latex]B'(x) = b^x B'(0)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Chain Rule Application:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^{g(x)}) = e^{g(x)} g'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169737923851\">Find the derivative of [latex]h(x)=xe^{2x}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"67723309\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"67723309\"]<\/p>\r\n<p id=\"fs-id1169738215078\">Don\u2019t forget to use the product rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737948362\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737948362\"]<\/p>\r\n<p id=\"fs-id1169737948362\">[latex]h^{\\prime}(x)=e^{2x}+2xe^{2x}[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=247&amp;end=287&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.9DerivativesOfExponentialAndLogarithmic247to287_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.9 Derivatives of Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = e^{\\tan(2x)}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"292328\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"292328\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]<br \/>\r\n\\begin{array}{rcl}<br \/>\r\nf'(x) &amp;=&amp; e^{\\tan(2x)} \\cdot \\frac{d}{dx}(\\tan(2x)) \\\\<br \/>\r\n&amp;=&amp; e^{\\tan(2x)} \\cdot \\sec^2(2x) \\cdot 2 \\\\<br \/>\r\n&amp;=&amp; 2e^{\\tan(2x)} \\sec^2(2x)<br \/>\r\n\\end{array}<br \/>\r\n[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738215955\">If [latex]A(t)=1000e^{0.3t}[\/latex] describes the mosquito population after [latex]t[\/latex] days, as in the preceding example, what is the rate of change of [latex]A(t)[\/latex] after 4 days?<\/p>\r\n<p>[reveal-answer q=\"883902\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"883902\"]<\/p>\r\n<p id=\"fs-id1169738225603\">Find [latex]A^{\\prime}(4)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737934408\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737934408\"]<\/p>\r\n<p id=\"fs-id1169737934408\">[latex]996[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Derivative of the Logarithmic Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Derivative of Natural Logarithm:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]x &gt; 0[\/latex], if [latex]y = \\ln x[\/latex], then [latex]\\frac{dy}{dx} = \\frac{1}{x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">General Logarithmic Derivative:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]h(x) = \\ln(g(x))[\/latex], [latex]h'(x) = \\frac{g'(x)}{g(x)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Derivative of General Base Logarithm:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]b &gt; 0, b \\neq 1[\/latex], if [latex]y = \\log_b x[\/latex], then [latex]\\frac{dy}{dx} = \\frac{1}{x \\ln b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Derivative of General Exponential:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]b &gt; 0, b \\neq 1[\/latex], if [latex]y = b^x[\/latex], then [latex]\\frac{dy}{dx} = b^x \\ln b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738219661\">Differentiate: [latex]f(x)=\\ln (3x+2)^5[\/latex].<\/p>\r\n<p>[reveal-answer q=\"644533\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"644533\"]<\/p>\r\n<p id=\"fs-id1169738073202\">Use a property of logarithms to simplify before taking the derivative.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738192196\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738192196\"]<\/p>\r\n<p id=\"fs-id1169738192196\">[latex]f^{\\prime}(x)=\\frac{15}{3x+2}[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=485&amp;end=523&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.9DerivativesOfExponentialAndLogarithmic485to523_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.9 Derivatives of Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = \\ln(\\frac{x^2 \\sin x}{2x+1})[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"229713\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"229713\"]<br \/>\r\nUse logarithm properties:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = \\ln(x^2) + \\ln(\\sin x) - \\ln(2x+1)[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Differentiate term by term:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]<br \/>\r\n\\begin{array}{rcl}<br \/>\r\nf'(x) &amp;=&amp; \\frac{2}{x} + \\frac{\\cos x}{\\sin x} - \\frac{2}{2x+1} \\\\<br \/>\r\n&amp;=&amp; \\frac{2}{x} + \\cot x - \\frac{2}{2x+1}<br \/>\r\n\\end{array}<br \/>\r\n[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the slope of the tangent line to [latex]y = \\log_2(3x+1)[\/latex] at [latex]x = 1[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"935504\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"935504\"]<\/p>\r\n<p>Use the general logarithm derivative formula:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx} = \\frac{3}{(3x+1)\\ln 2}[\/latex]<\/p>\r\n<p>Evaluate at [latex]x = 1[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx}\\Big|_{x=1} = \\frac{3}{4\\ln 2} = \\frac{3}{\\ln 16}[\/latex]<\/p>\r\n<p>The slope of the tangent line at [latex]x = 1[\/latex] is [latex]\\frac{3}{\\ln 16}[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738240290\">Find the slope for the line tangent to [latex]y=3^x[\/latex] at [latex]x=2[\/latex].<\/p>\r\n<p>[reveal-answer q=\"299031\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"299031\"]<\/p>\r\n<p id=\"fs-id1169737934313\">Evaluate the derivative at [latex]x=2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737934288\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737934288\"]<\/p>\r\n<p id=\"fs-id1169737934288\">[latex]9 \\ln (3)[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=756&amp;end=790&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.9DerivativesOfExponentialAndLogarithmic756to790_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.9 Derivatives of Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Logarithmic Differentiation<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Purpose:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Simplify differentiation of complex functions, especially those involving products, quotients, and exponents<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Functions of the form [latex]y = (g(x))^n[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Functions like [latex]y = b^{g(x)}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Expressions such as [latex]y = x^x[\/latex] or [latex]y = x^{\\pi}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-bold\"><strong>Step-by-Step Process<\/strong><\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Take natural logarithm of both sides: [latex]\\ln y = \\ln(h(x))[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Expand using logarithm properties<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Differentiate both sides implicitly<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplify the final expression<\/li>\r\n<\/ol>\r\n<p class=\"font-bold\"><strong>Key Logarithm Properties<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Product Rule: [latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Quotient Rule: [latex]\\log_b(\\frac{M}{N}) = \\log_b(M) - \\log_b(N)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power Rule: [latex]\\log_b(M^n) = n\\log_b(M)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Change of Base: [latex]\\log_b(M) = \\frac{\\log_n(M)}{\\log_n(b)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738068356\">Find the derivative of [latex]y=\\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738201884\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738201884\"]This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter.<\/p>\r\n<div id=\"fs-id1169738201891\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} \\ln y &amp; = \\ln \\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x} &amp; &amp; &amp; \\text{Step 1. Take the natural logarithm of both sides.} \\\\ \\ln y &amp; = \\ln x+\\frac{1}{2} \\ln (2x+1)-x \\ln e-3 \\ln \\sin x &amp; &amp; &amp; \\text{Step 2. Expand using properties of logarithms.} \\\\ \\frac{1}{y}\\frac{dy}{dx} &amp; = \\frac{1}{x}+\\frac{1}{2x+1}-1-3\\big(\\frac{\\cos x}{\\sin x}\\big) &amp; &amp; &amp; \\text{Step 3. Differentiate both sides.} \\\\ \\frac{dy}{dx} &amp; = y (\\frac{1}{x}+\\frac{1}{2x+1}-1-3 \\cot x) &amp; &amp; &amp; \\text{Step 4. Multiply by} \\, y \\, \\text{on both sides and simplify.} \\\\ \\frac{dy}{dx} &amp; = \\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x} \\normalsize (\\frac{1}{x}+\\frac{1}{2x+1}-1-3 \\cot x) &amp; &amp; &amp; \\text{Step 5. Substitute} \\, y=\\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x}. \\end{array}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\">[\/hidden-answer]<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169737933517\">Use logarithmic differentiation to find the derivative of [latex]y=x^x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737933537\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737933537\"]<\/p>\r\n<p id=\"fs-id1169737933537\">[latex]\\frac{dy}{dx}=x^x(1+\\ln x)[\/latex]<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=1160&amp;end=1223&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.9DerivativesOfExponentialAndLogarithmic1160to1223_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.9 Derivatives of Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169737933602\">Find the derivative of [latex]y=(\\tan x)^{\\pi}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169738233543\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738233543\"]<\/p>\r\n<p id=\"fs-id1169738233543\">[latex]y^{\\prime}=\\pi (\\tan x)^{\\pi -1} \\sec^2 x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the derivative of [latex]y = (2x^4 + 1)^{\\tan x}[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"828783\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"828783\"]<\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Take natural log:<center>[latex]\\ln y = \\ln(2x^4 + 1)^{\\tan x}[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply log power rule:<center>[latex]\\ln y = \\tan x \\ln(2x^4 + 1)[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Differentiate both sides:<center>[latex]\\frac{1}{y}\\frac{dy}{dx} = \\sec^2 x \\ln(2x^4 + 1) + \\tan x \\cdot \\dfrac{8x^3}{2x^4 + 1}[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex]:<center>[latex]\\frac{dy}{dx} = y(\\sec^2 x \\ln(2x^4 + 1) + \\tan x \\cdot \\dfrac{8x^3}{2x^4 + 1})[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Substitute [latex]y[\/latex]:<center>[latex]\\frac{dy}{dx} = (2x^4 + 1)^{\\tan x}(\\sec^2 x \\ln(2x^4 + 1) + \\tan x \\cdot \\dfrac{8x^3}{2x^4 + 1})[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]y = x^r[\/latex] where [latex]r[\/latex] is any real number.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"798382\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"798382\"]<\/p>\r\n<ol>\r\n\t<li class=\"whitespace-normal break-words\">Take natural log:<center>[latex]\\ln y = \\ln x^r[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply log power rule:<center>[latex]\\ln y = r \\ln x[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Differentiate:<center>[latex]\\frac{1}{y}\\frac{dy}{dx} = r \\cdot \\frac{1}{x}[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex]:<center>\u00a0[latex]\\frac{dy}{dx} = y \\cdot \\frac{r}{x}[\/latex]<\/center><\/li>\r\n\t<li class=\"whitespace-normal break-words\">Substitute [latex]y[\/latex] and simplify:<center>[latex]\\frac{dy}{dx} = rx^{r-1}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine the derivatives of exponential and logarithmic functions<\/li>\n<li>Apply logarithmic differentiation to find derivatives<\/li>\n<\/ul>\n<\/section>\n<h2>Derivative of the Exponential Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">The Natural Exponential Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Defined as [latex]E(x) = e^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e \\approx 2.718281828...[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^x) = e^x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Exponential Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]B(x) = b^x[\/latex] where [latex]b > 0[\/latex]: [latex]B'(x) = b^x B'(0)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Chain Rule Application:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^{g(x)}) = e^{g(x)} g'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169737923851\">Find the derivative of [latex]h(x)=xe^{2x}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q67723309\">Hint<\/button><\/p>\n<div id=\"q67723309\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738215078\">Don\u2019t forget to use the product rule.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737948362\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737948362\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737948362\">[latex]h^{\\prime}(x)=e^{2x}+2xe^{2x}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=247&amp;end=287&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.9DerivativesOfExponentialAndLogarithmic247to287_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.9 Derivatives of Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = e^{\\tan(2x)}[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q292328\">Show Answer<\/button><\/p>\n<div id=\"q292328\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]<br \/>  \\begin{array}{rcl}<br \/>  f'(x) &=& e^{\\tan(2x)} \\cdot \\frac{d}{dx}(\\tan(2x)) \\\\<br \/>  &=& e^{\\tan(2x)} \\cdot \\sec^2(2x) \\cdot 2 \\\\<br \/>  &=& 2e^{\\tan(2x)} \\sec^2(2x)<br \/>  \\end{array}<br \/>[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738215955\">If [latex]A(t)=1000e^{0.3t}[\/latex] describes the mosquito population after [latex]t[\/latex] days, as in the preceding example, what is the rate of change of [latex]A(t)[\/latex] after 4 days?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q883902\">Hint<\/button><\/p>\n<div id=\"q883902\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738225603\">Find [latex]A^{\\prime}(4)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737934408\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737934408\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737934408\">[latex]996[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Derivative of the Logarithmic Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Derivative of Natural Logarithm:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]x > 0[\/latex], if [latex]y = \\ln x[\/latex], then [latex]\\frac{dy}{dx} = \\frac{1}{x}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Logarithmic Derivative:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]h(x) = \\ln(g(x))[\/latex], [latex]h'(x) = \\frac{g'(x)}{g(x)}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivative of General Base Logarithm:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]b > 0, b \\neq 1[\/latex], if [latex]y = \\log_b x[\/latex], then [latex]\\frac{dy}{dx} = \\frac{1}{x \\ln b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivative of General Exponential:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]b > 0, b \\neq 1[\/latex], if [latex]y = b^x[\/latex], then [latex]\\frac{dy}{dx} = b^x \\ln b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738219661\">Differentiate: [latex]f(x)=\\ln (3x+2)^5[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q644533\">Hint<\/button><\/p>\n<div id=\"q644533\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738073202\">Use a property of logarithms to simplify before taking the derivative.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738192196\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738192196\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738192196\">[latex]f^{\\prime}(x)=\\frac{15}{3x+2}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=485&amp;end=523&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.9DerivativesOfExponentialAndLogarithmic485to523_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.9 Derivatives of Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]f(x) = \\ln(\\frac{x^2 \\sin x}{2x+1})[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q229713\">Show Answer<\/button><\/p>\n<div id=\"q229713\" class=\"hidden-answer\" style=\"display: none\">\nUse logarithm properties:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = \\ln(x^2) + \\ln(\\sin x) - \\ln(2x+1)[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Differentiate term by term:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]<br \/>  \\begin{array}{rcl}<br \/>  f'(x) &=& \\frac{2}{x} + \\frac{\\cos x}{\\sin x} - \\frac{2}{2x+1} \\\\<br \/>  &=& \\frac{2}{x} + \\cot x - \\frac{2}{2x+1}<br \/>  \\end{array}<br \/>[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the slope of the tangent line to [latex]y = \\log_2(3x+1)[\/latex] at [latex]x = 1[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q935504\">Show Answer<\/button><\/p>\n<div id=\"q935504\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the general logarithm derivative formula:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx} = \\frac{3}{(3x+1)\\ln 2}[\/latex]<\/p>\n<p>Evaluate at [latex]x = 1[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{dy}{dx}\\Big|_{x=1} = \\frac{3}{4\\ln 2} = \\frac{3}{\\ln 16}[\/latex]<\/p>\n<p>The slope of the tangent line at [latex]x = 1[\/latex] is [latex]\\frac{3}{\\ln 16}[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738240290\">Find the slope for the line tangent to [latex]y=3^x[\/latex] at [latex]x=2[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q299031\">Hint<\/button><\/p>\n<div id=\"q299031\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737934313\">Evaluate the derivative at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737934288\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737934288\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737934288\">[latex]9 \\ln (3)[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=756&amp;end=790&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\/3.9DerivativesOfExponentialAndLogarithmic756to790_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.9 Derivatives of Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Logarithmic Differentiation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplify differentiation of complex functions, especially those involving products, quotients, and exponents<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Functions of the form [latex]y = (g(x))^n[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Functions like [latex]y = b^{g(x)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Expressions such as [latex]y = x^x[\/latex] or [latex]y = x^{\\pi}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Step-by-Step Process<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Take natural logarithm of both sides: [latex]\\ln y = \\ln(h(x))[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Expand using logarithm properties<\/li>\n<li class=\"whitespace-normal break-words\">Differentiate both sides implicitly<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify the final expression<\/li>\n<\/ol>\n<p class=\"font-bold\"><strong>Key Logarithm Properties<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Product Rule: [latex]\\log_b(MN) = \\log_b(M) + \\log_b(N)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quotient Rule: [latex]\\log_b(\\frac{M}{N}) = \\log_b(M) - \\log_b(N)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Power Rule: [latex]\\log_b(M^n) = n\\log_b(M)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Change of Base: [latex]\\log_b(M) = \\frac{\\log_n(M)}{\\log_n(b)}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738068356\">Find the derivative of [latex]y=\\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738201884\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738201884\" class=\"hidden-answer\" style=\"display: none\">This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter.<\/p>\n<div id=\"fs-id1169738201891\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} \\ln y & = \\ln \\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x} & & & \\text{Step 1. Take the natural logarithm of both sides.} \\\\ \\ln y & = \\ln x+\\frac{1}{2} \\ln (2x+1)-x \\ln e-3 \\ln \\sin x & & & \\text{Step 2. Expand using properties of logarithms.} \\\\ \\frac{1}{y}\\frac{dy}{dx} & = \\frac{1}{x}+\\frac{1}{2x+1}-1-3\\big(\\frac{\\cos x}{\\sin x}\\big) & & & \\text{Step 3. Differentiate both sides.} \\\\ \\frac{dy}{dx} & = y (\\frac{1}{x}+\\frac{1}{2x+1}-1-3 \\cot x) & & & \\text{Step 4. Multiply by} \\, y \\, \\text{on both sides and simplify.} \\\\ \\frac{dy}{dx} & = \\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x} \\normalsize (\\frac{1}{x}+\\frac{1}{2x+1}-1-3 \\cot x) & & & \\text{Step 5. Substitute} \\, y=\\large \\frac{x\\sqrt{2x+1}}{e^x \\sin^3 x}. \\end{array}[\/latex]<\/div>\n<div class=\"equation unnumbered\"><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169737933517\">Use logarithmic differentiation to find the derivative of [latex]y=x^x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737933537\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737933537\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737933537\">[latex]\\frac{dy}{dx}=x^x(1+\\ln x)[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_nzxTKiFPpo?controls=0&amp;start=1160&amp;end=1223&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\/3.9DerivativesOfExponentialAndLogarithmic1160to1223_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.9 Derivatives of Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169737933602\">Find the derivative of [latex]y=(\\tan x)^{\\pi}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738233543\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738233543\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738233543\">[latex]y^{\\prime}=\\pi (\\tan x)^{\\pi -1} \\sec^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the derivative of [latex]y = (2x^4 + 1)^{\\tan x}[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q828783\">Show Answer<\/button><\/p>\n<div id=\"q828783\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Take natural log:\n<div style=\"text-align: center;\">[latex]\\ln y = \\ln(2x^4 + 1)^{\\tan x}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Apply log power rule:\n<div style=\"text-align: center;\">[latex]\\ln y = \\tan x \\ln(2x^4 + 1)[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Differentiate both sides:\n<div style=\"text-align: center;\">[latex]\\frac{1}{y}\\frac{dy}{dx} = \\sec^2 x \\ln(2x^4 + 1) + \\tan x \\cdot \\dfrac{8x^3}{2x^4 + 1}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex]:\n<div style=\"text-align: center;\">[latex]\\frac{dy}{dx} = y(\\sec^2 x \\ln(2x^4 + 1) + \\tan x \\cdot \\dfrac{8x^3}{2x^4 + 1})[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]y[\/latex]:\n<div style=\"text-align: center;\">[latex]\\frac{dy}{dx} = (2x^4 + 1)^{\\tan x}(\\sec^2 x \\ln(2x^4 + 1) + \\tan x \\cdot \\dfrac{8x^3}{2x^4 + 1})[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the derivative of [latex]y = x^r[\/latex] where [latex]r[\/latex] is any real number.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q798382\">Show Answer<\/button><\/p>\n<div id=\"q798382\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Take natural log:\n<div style=\"text-align: center;\">[latex]\\ln y = \\ln x^r[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Apply log power rule:\n<div style=\"text-align: center;\">[latex]\\ln y = r \\ln x[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Differentiate:\n<div style=\"text-align: center;\">[latex]\\frac{1}{y}\\frac{dy}{dx} = r \\cdot \\frac{1}{x}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]\\frac{dy}{dx}[\/latex]:\n<div style=\"text-align: center;\">\u00a0[latex]\\frac{dy}{dx} = y \\cdot \\frac{r}{x}[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]y[\/latex] and simplify:\n<div style=\"text-align: center;\">[latex]\\frac{dy}{dx} = rx^{r-1}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap 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