{"id":613,"date":"2025-10-09T15:25:34","date_gmt":"2025-10-09T15:25:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=613"},"modified":"2025-11-20T15:58:03","modified_gmt":"2025-11-20T15:58:03","slug":"advanced-partial-derivatives-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/advanced-partial-derivatives-background-youll-need-1\/","title":{"raw":"Advanced Partial Derivatives: Background You'll Need 1","rendered":"Advanced Partial Derivatives: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Differentiate exponential, logarithmic, and trigonometric functions <\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"font-claude-response-heading text-text-100 mt-1 -mb-0.5\">Derivatives of Exponential, Logarithmic, and Trigonometric Functions<\/h2>\r\n<h3 class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\">The Natural Exponential Function<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The function [latex]f(x) = e^x[\/latex] has a unique property: it's its own derivative.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>derivative of [latex]e^x[\/latex]<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(e^x) = e^x[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally, using the chain rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(e^{g(x)}) = e^{g(x)} \\cdot g'(x)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Think of it as: [latex]e[\/latex] raised to the power of a function, times the derivative of that function.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1169737766547\">Find the derivative of [latex]y=\\dfrac{e^{x^2}}{x}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169737928258\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737928258\"]\r\n<p id=\"fs-id1169737928258\">Use the derivative of the natural exponential function, the quotient rule, and the chain rule.<\/p>\r\n\r\n<div id=\"fs-id1169737928262\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} y^{\\prime} &amp; =\\large \\frac{(e^{x^2} \\cdot 2x) \\cdot x - 1 \\cdot e^{x^2}}{x^2} &amp; &amp; &amp; \\text{Apply the quotient rule.} \\\\ &amp; = \\large \\frac{e^{x^2}(2x^2-1)}{x^2} &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]33753[\/ohm_question]\r\n\r\n<\/section>\r\n<h3 class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\">General Exponential Functions<\/h3>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>derivative of [latex]b^x[\/latex]<\/h3>\r\nFor exponential functions with base [latex]b \\neq e[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(b^x) = b^x \\ln b[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(b^{g(x)}) = b^{g(x)} \\cdot g'(x) \\cdot \\ln b[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The extra factor of [latex]\\ln b[\/latex] accounts for the base being different from [latex]e[\/latex].<\/p>\r\n\r\n<\/section>\r\n<h3>The Natural Logarithm<\/h3>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>derivative of [latex]\\ln x[\/latex]<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For [latex]x &gt; 0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\ln x) = \\dfrac{1}{x}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally, using the chain rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\ln(g(x))) = \\dfrac{g'(x)}{g(x)}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">This applies when [latex]g(x) &gt; 0[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1169738228377\">Find the derivative of [latex]f(x)=\\ln(x^3+3x-4)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169738221312\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738221312\"]\r\n<p id=\"fs-id1169738221312\">Use the derivative of a natural logarithm directly.<\/p>\r\n\r\n<div id=\"fs-id1169738220266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} f^{\\prime}(x) &amp; =\\frac{1}{x^3+3x-4} \\cdot (3x^2+3) &amp; &amp; &amp; \\text{Use} \\, g(x)=x^3+3x-4 \\, \\text{in} \\, h^{\\prime}(x)=\\frac{1}{g(x)} g^{\\prime}(x). \\\\ &amp; =\\frac{3x^2+3}{x^3+3x-4} &amp; &amp; &amp; \\text{Rewrite.} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3>General Logarithmic Functions<\/h3>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>derivative of [latex]\\log_b x[\/latex]<\/h3>\r\nFor logarithms with base [latex]b \\neq e[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\log_b x) = \\dfrac{1}{x \\ln b}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\log_b(g(x))) = \\dfrac{g'(x)}{g(x) \\ln b}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Again, the factor of [latex]\\ln b[\/latex] adjusts for the different base.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1169738185259\">Find the derivative of [latex]h(x)= \\dfrac{3^x}{3^x+2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169737700313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169737700313\"]\r\n<p id=\"fs-id1169737700313\">Use the quotient rule and the derivative from above.<\/p>\r\n\r\n<div id=\"fs-id1169737700320\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} h^{\\prime}(x) &amp; = \\large \\frac{3^x \\ln 3(3^x+2)-3^x \\ln 3(3^x)}{(3^x+2)^2} &amp; &amp; &amp; \\text{Apply the quotient rule.} \\\\ &amp; = \\large \\frac{2 \\cdot 3^x \\ln 3}{(3^x+2)^2} &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288388[\/ohm_question]\r\n\r\n<\/section>\r\n<h3>Basic Trigonometric Derivatives<\/h3>\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>derivatives of sine and cosine<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\sin x) = \\cos x[\/latex]&lt;\/center&gt; &lt;center&gt;[latex]\\dfrac{d}{dx}(\\cos x) = -\\sin x[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Notice the negative sign for cosine\u2014this reflects the relationship between the functions and their graphs.<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Understanding the Sign Pattern<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Watch the signs carefully:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sin x[\/latex] differentiates to [latex]+\\cos x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\cos x[\/latex] differentiates to [latex]-\\sin x[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">At points where [latex]\\sin x[\/latex] has a horizontal tangent (maximum or minimum), [latex]\\cos x = 0[\/latex]. Where [latex]\\sin x[\/latex] is increasing, [latex]\\cos x &gt; 0[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5x^3 \\sin x[\/latex].<\/p>\r\n[reveal-answer q=\"300277\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"300277\"]\r\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739028319\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739028319\"]Using the product rule, we have\r\n<div id=\"fs-id1169739001004\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) &amp; =\\frac{d}{dx}(5x^3)\\cdot \\sin x+\\frac{d}{dx}(\\sin x)\\cdot 5x^3 \\\\ &amp; =15x^2\\cdot \\sin x+ \\cos x\\cdot 5x^3\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739269866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=15x^2 \\sin x+5x^3 \\cos x[\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\dfrac{\\cos x}{4x^2}[\/latex].<\/p>\r\n[reveal-answer q=\"488399\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"488399\"]\r\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169738969705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738969705\"]\r\n<p id=\"fs-id1169738969705\">By applying the quotient rule, we have<\/p>\r\n\r\n<div id=\"fs-id1169738960497\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\frac{(\u2212\\sin x)4x^2-8x(\\cos x)}{(4x^2)^2}[\/latex].<\/div>\r\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739179211\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}g^{\\prime}(x) &amp; =\\frac{-4x^2 \\sin x-8x \\cos x}{16x^4} \\\\ &amp; =\\frac{\u2212x \\sin x-2 \\cos x}{4x^3} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]205604[\/ohm_question]\r\n\r\n<\/section>\r\n<h3 class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\">Other Trigonometric Functions<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The remaining trigonometric functions can be differentiated using the quotient rule, since they're defined as ratios of sine and cosine.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>derivatives of all six trigonometric functions<\/h3>\r\n[latex]\\dfrac{d}{dx}(\\sin x) = \\cos x[\/latex]\r\n[latex]\\dfrac{d}{dx}(\\cos x) = -\\sin x[\/latex]\r\n[latex]\\dfrac{d}{dx}(\\tan x) = \\sec^2 x[\/latex]\r\n[latex]\\dfrac{d}{dx}(\\cot x) = -\\csc^2 x[\/latex]\r\n[latex]\\dfrac{d}{dx}(\\sec x) = \\sec x \\tan x[\/latex]\r\n[latex]\\dfrac{d}{dx}(\\csc x) = -\\csc x \\cot x[\/latex]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Pattern Recognition<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Notice the patterns:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Functions starting with \"co-\" have negative derivatives<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Squared functions ([latex]\\sec^2 x[\/latex], [latex]\\csc^2 x[\/latex]) come from [latex]\\tan x[\/latex] and [latex]\\cot x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Products ([latex]\\sec x \\tan x[\/latex], [latex]\\csc x \\cot x[\/latex]) come from [latex]\\sec x[\/latex] and [latex]\\csc x[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1169736655855\">Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736655897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736655897\"]\r\n<p id=\"fs-id1169736655897\">To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find<\/p>\r\n\r\n<div id=\"fs-id1169736610100\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\frac{d}{dx}(\\csc x)+\\frac{d}{dx}(x \\tan x)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169736610172\">In the first term, [latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex], and by applying the product rule to the second term we obtain<\/p>\r\n\r\n<div id=\"fs-id1169739182390\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(x \\tan x)=(1)(\\tan x)+(\\sec^2 x)(x)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739265934\">Therefore, we have<\/p>\r\n\r\n<div id=\"fs-id1169739265938\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\u2212\\csc x \\cot x+ \\tan x+x \\sec^2 x[\/latex].<\/div>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1247&amp;end=1311&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.5DerivativesofTrigonometricFunctions1247to1311_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5 Derivatives of Trigonometric Functions (edited)\" here (opens in new window)<\/a>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]33737[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Differentiate exponential, logarithmic, and trigonometric functions <\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"font-claude-response-heading text-text-100 mt-1 -mb-0.5\">Derivatives of Exponential, Logarithmic, and Trigonometric Functions<\/h2>\n<h3 class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\">The Natural Exponential Function<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The function [latex]f(x) = e^x[\/latex] has a unique property: it&#8217;s its own derivative.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>derivative of [latex]e^x[\/latex]<\/h3>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(e^x) = e^x[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally, using the chain rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(e^{g(x)}) = e^{g(x)} \\cdot g'(x)[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Think of it as: [latex]e[\/latex] raised to the power of a function, times the derivative of that function.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1169737766547\">Find the derivative of [latex]y=\\dfrac{e^{x^2}}{x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737928258\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737928258\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737928258\">Use the derivative of the natural exponential function, the quotient rule, and the chain rule.<\/p>\n<div id=\"fs-id1169737928262\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} y^{\\prime} & =\\large \\frac{(e^{x^2} \\cdot 2x) \\cdot x - 1 \\cdot e^{x^2}}{x^2} & & & \\text{Apply the quotient rule.} \\\\ & = \\large \\frac{e^{x^2}(2x^2-1)}{x^2} & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm33753\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33753&theme=lumen&iframe_resize_id=ohm33753&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3 class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\">General Exponential Functions<\/h3>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>derivative of [latex]b^x[\/latex]<\/h3>\n<p>For exponential functions with base [latex]b \\neq e[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(b^x) = b^x \\ln b[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(b^{g(x)}) = b^{g(x)} \\cdot g'(x) \\cdot \\ln b[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The extra factor of [latex]\\ln b[\/latex] accounts for the base being different from [latex]e[\/latex].<\/p>\n<\/section>\n<h3>The Natural Logarithm<\/h3>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>derivative of [latex]\\ln x[\/latex]<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For [latex]x > 0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\ln x) = \\dfrac{1}{x}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally, using the chain rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\ln(g(x))) = \\dfrac{g'(x)}{g(x)}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">This applies when [latex]g(x) > 0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1169738228377\">Find the derivative of [latex]f(x)=\\ln(x^3+3x-4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738221312\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738221312\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738221312\">Use the derivative of a natural logarithm directly.<\/p>\n<div id=\"fs-id1169738220266\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} f^{\\prime}(x) & =\\frac{1}{x^3+3x-4} \\cdot (3x^2+3) & & & \\text{Use} \\, g(x)=x^3+3x-4 \\, \\text{in} \\, h^{\\prime}(x)=\\frac{1}{g(x)} g^{\\prime}(x). \\\\ & =\\frac{3x^2+3}{x^3+3x-4} & & & \\text{Rewrite.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>General Logarithmic Functions<\/h3>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>derivative of [latex]\\log_b x[\/latex]<\/h3>\n<p>For logarithms with base [latex]b \\neq e[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\log_b x) = \\dfrac{1}{x \\ln b}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">More generally:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\log_b(g(x))) = \\dfrac{g'(x)}{g(x) \\ln b}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Again, the factor of [latex]\\ln b[\/latex] adjusts for the different base.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1169738185259\">Find the derivative of [latex]h(x)= \\dfrac{3^x}{3^x+2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737700313\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737700313\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737700313\">Use the quotient rule and the derivative from above.<\/p>\n<div id=\"fs-id1169737700320\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll} h^{\\prime}(x) & = \\large \\frac{3^x \\ln 3(3^x+2)-3^x \\ln 3(3^x)}{(3^x+2)^2} & & & \\text{Apply the quotient rule.} \\\\ & = \\large \\frac{2 \\cdot 3^x \\ln 3}{(3^x+2)^2} & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288388\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288388&theme=lumen&iframe_resize_id=ohm288388&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3>Basic Trigonometric Derivatives<\/h3>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>derivatives of sine and cosine<\/h3>\n<p style=\"text-align: center;\">[latex]\\dfrac{d}{dx}(\\sin x) = \\cos x[\/latex]&lt;\/center&gt; &lt;center&gt;[latex]\\dfrac{d}{dx}(\\cos x) = -\\sin x[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Notice the negative sign for cosine\u2014this reflects the relationship between the functions and their graphs.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Understanding the Sign Pattern<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Watch the signs carefully:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\sin x[\/latex] differentiates to [latex]+\\cos x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\cos x[\/latex] differentiates to [latex]-\\sin x[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body whitespace-normal break-words\">At points where [latex]\\sin x[\/latex] has a horizontal tangent (maximum or minimum), [latex]\\cos x = 0[\/latex]. Where [latex]\\sin x[\/latex] is increasing, [latex]\\cos x > 0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1169739269454\">Find the derivative of [latex]f(x)=5x^3 \\sin x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q300277\">Hint<\/button><\/p>\n<div id=\"q300277\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738821957\">Don\u2019t forget to use the product rule.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169739028319\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169739028319\" class=\"hidden-answer\" style=\"display: none\">Using the product rule, we have<\/p>\n<div id=\"fs-id1169739001004\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}f^{\\prime}(x) & =\\frac{d}{dx}(5x^3)\\cdot \\sin x+\\frac{d}{dx}(\\sin x)\\cdot 5x^3 \\\\ & =15x^2\\cdot \\sin x+ \\cos x\\cdot 5x^3\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169739036358\">After simplifying, we obtain<\/p>\n<div id=\"fs-id1169739269866\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=15x^2 \\sin x+5x^3 \\cos x[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1169739229519\">Find the derivative of [latex]g(x)=\\dfrac{\\cos x}{4x^2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q488399\">Hint<\/button><\/p>\n<div id=\"q488399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736587923\">Use the quotient rule.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738969705\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738969705\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738969705\">By applying the quotient rule, we have<\/p>\n<div id=\"fs-id1169738960497\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g^{\\prime}(x)=\\frac{(\u2212\\sin x)4x^2-8x(\\cos x)}{(4x^2)^2}[\/latex].<\/div>\n<p id=\"fs-id1169738949562\">Simplifying, we obtain<\/p>\n<div id=\"fs-id1169739179211\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll}g^{\\prime}(x) & =\\frac{-4x^2 \\sin x-8x \\cos x}{16x^4} \\\\ & =\\frac{\u2212x \\sin x-2 \\cos x}{4x^3} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm205604\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=205604&theme=lumen&iframe_resize_id=ohm205604&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<h3 class=\"font-claude-response-subheading text-text-100 mt-1 -mb-1.5\">Other Trigonometric Functions<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The remaining trigonometric functions can be differentiated using the quotient rule, since they&#8217;re defined as ratios of sine and cosine.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>derivatives of all six trigonometric functions<\/h3>\n<p>[latex]\\dfrac{d}{dx}(\\sin x) = \\cos x[\/latex]<br \/>\n[latex]\\dfrac{d}{dx}(\\cos x) = -\\sin x[\/latex]<br \/>\n[latex]\\dfrac{d}{dx}(\\tan x) = \\sec^2 x[\/latex]<br \/>\n[latex]\\dfrac{d}{dx}(\\cot x) = -\\csc^2 x[\/latex]<br \/>\n[latex]\\dfrac{d}{dx}(\\sec x) = \\sec x \\tan x[\/latex]<br \/>\n[latex]\\dfrac{d}{dx}(\\csc x) = -\\csc x \\cot x[\/latex]<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Pattern Recognition<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Notice the patterns:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Functions starting with &#8220;co-&#8221; have negative derivatives<\/li>\n<li class=\"whitespace-normal break-words\">Squared functions ([latex]\\sec^2 x[\/latex], [latex]\\csc^2 x[\/latex]) come from [latex]\\tan x[\/latex] and [latex]\\cot x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Products ([latex]\\sec x \\tan x[\/latex], [latex]\\csc x \\cot x[\/latex]) come from [latex]\\sec x[\/latex] and [latex]\\csc x[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1169736655855\">Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169736655897\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169736655897\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736655897\">To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find<\/p>\n<div id=\"fs-id1169736610100\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\frac{d}{dx}(\\csc x)+\\frac{d}{dx}(x \\tan x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169736610172\">In the first term, [latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex], and by applying the product rule to the second term we obtain<\/p>\n<div id=\"fs-id1169739182390\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx}(x \\tan x)=(1)(\\tan x)+(\\sec^2 x)(x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739265934\">Therefore, we have<\/p>\n<div id=\"fs-id1169739265938\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\u2212\\csc x \\cot x+ \\tan x+x \\sec^2 x[\/latex].<\/div>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hvsQJFir7Qw?controls=0&amp;start=1247&amp;end=1311&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\/3.5DerivativesofTrigonometricFunctions1247to1311_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5 Derivatives of Trigonometric Functions (edited)&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm33737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33737&theme=lumen&iframe_resize_id=ohm33737&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-613","chapter","type-chapter","status-publish","hentry"],"part":428,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/613","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/613\/revisions"}],"predecessor-version":[{"id":2641,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/613\/revisions\/2641"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/428"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/613\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=613"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=613"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=613"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=613"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}