{"id":3595,"date":"2024-06-28T17:12:22","date_gmt":"2024-06-28T17:12:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3595"},"modified":"2024-08-05T01:53:42","modified_gmt":"2024-08-05T01:53:42","slug":"techniques-for-differentiation-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/techniques-for-differentiation-cheat-sheet\/","title":{"raw":"Techniques for Differentiation: Cheat Sheet","rendered":"Techniques for Differentiation: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Techniques+for+Differentiation.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<p><strong>Derivatives of Trigonometric Functions<\/strong><\/p>\r\n<ul id=\"fs-id1169739325602\">\r\n\t<li>We can find the derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] by using the definition of derivative and the limit formulas found earlier. The results are<br \/>\r\n<div id=\"fs-id1169736589261\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx} \\sin x= \\cos x[\/latex]\u00a0 and\u00a0 [latex]\\frac{d}{dx} \\cos x=\u2212\\sin x[\/latex].<\/div>\r\n<\/li>\r\n\t<li>With these two formulas, we can determine the derivatives of all six basic trigonometric functions.<\/li>\r\n<\/ul>\r\n<p><strong>The Chain Rule<\/strong><\/p>\r\n<ul id=\"fs-id1169736594108\">\r\n\t<li>The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x))[\/latex],<br \/>\r\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/div>\r\n<ul>\r\n\t<li>\r\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: left;\">In Leibniz\u2019s notation this rule takes the form<\/div>\r\n<div id=\"fs-id1169737159954\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.<\/li>\r\n\t<li>The chain rule combines with the power rule to form a new rule:<br \/>\r\n<div id=\"fs-id1169737470447\" class=\"equation unnumbered\" style=\"text-align: center;\">If [latex]h(x)=(g(x))^n[\/latex], then [latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\r\n<\/li>\r\n\t<li style=\"text-align: left;\">When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x)))[\/latex], then [latex]h^{\\prime}(x)=f^{\\prime}(g(k(x)))g^{\\prime}(k(x))k^{\\prime}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives of Inverse Functions<\/strong><\/p>\r\n<ul id=\"fs-id1169736611730\">\r\n\t<li>The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.<\/li>\r\n\t<li>We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.<\/li>\r\n<\/ul>\r\n<p><strong>Implicit Differentiation<\/strong><\/p>\r\n<ul id=\"fs-id1169738240218\">\r\n\t<li>We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).<\/li>\r\n\t<li>By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives of Exponential and Logarithmic Functions<\/strong><\/p>\r\n<ul id=\"fs-id1169738233616\">\r\n\t<li>On the basis of the assumption that the exponential function [latex]y=b^x, \\, b&gt;0[\/latex] is continuous everywhere and differentiable at [latex]0[\/latex], this function is differentiable everywhere and there is a formula for its derivative.<\/li>\r\n\t<li>We can use a formula to find the derivative of [latex]y=\\ln x[\/latex], and the relationship [latex]\\log_b x=\\dfrac{\\ln x}{\\ln b}[\/latex] allows us to extend our differentiation formulas to include logarithms with arbitrary bases.<\/li>\r\n\t<li>Logarithmic differentiation allows us to differentiate functions of the form [latex]y=g(x)^{f(x)}[\/latex] or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n\t<li><strong>Derivative of sine function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/li>\r\n\t<li><strong>Derivative of cosine function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/li>\r\n\t<li><strong>Derivative of tangent function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/li>\r\n\t<li><strong>Derivative of cotangent function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/li>\r\n\t<li><strong>Derivative of secant function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/li>\r\n\t<li><strong>Derivative of cosecant function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/li>\r\n\t<li><strong>The chain rule<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(f(g(x)))=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/li>\r\n\t<li><strong>The power rule for functions<\/strong><br \/>\r\n[latex]\\frac{d}{dx}((g(x)^n)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/li>\r\n\t<li><strong>Derivative of inverse sine function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\sin^{-1} x)=\\dfrac{1}{\\sqrt{1-x^2}}[\/latex]<\/li>\r\n\t<li><strong>Derivative of inverse cosine function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\cos^{-1} x)=\\dfrac{-1}{\\sqrt{1-x^2}}[\/latex]<\/li>\r\n\t<li><strong>Derivative of inverse tangent function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\tan^{-1} x)=\\dfrac{1}{1+x^2}[\/latex]<\/li>\r\n\t<li><strong>Derivative of inverse cotangent function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\cot^{-1} x)=\\dfrac{-1}{1+x^2}[\/latex]<\/li>\r\n\t<li><strong>Derivative of inverse secant function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\sec^{-1} x)=\\dfrac{1}{|x|\\sqrt{x^2-1}}[\/latex]<\/li>\r\n\t<li><strong>Derivative of inverse cosecant function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\csc^{-1} x)=\\dfrac{-1}{|x|\\sqrt{x^2-1}}[\/latex]<\/li>\r\n\t<li><strong>Inverse function theorem<\/strong><br \/>\r\n[latex](f^{-1})^{\\prime}(x)=\\dfrac{1}{f^{\\prime}(f^{-1}(x))}[\/latex] whenever [latex]f^{\\prime}(f^{-1}(x))\\ne 0[\/latex] and [latex]f(x)[\/latex] is differentiable.<\/li>\r\n\t<li><strong>Power rule with rational exponents<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(x^{m\/n})=\\frac{m}{n}x^{(m\/n)-1}[\/latex].<\/li>\r\n\t<li><strong>Derivative of the natural exponential function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\\prime}(x)[\/latex]<\/li>\r\n\t<li><strong>Derivative of the natural logarithmic function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\ln (g(x)))=\\dfrac{1}{g(x)} g^{\\prime}(x)[\/latex]<\/li>\r\n\t<li><strong>Derivative of the general exponential function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(b^{g(x)})=b^{g(x)} g^{\\prime}(x) \\ln b[\/latex]<\/li>\r\n\t<li><strong>Derivative of the general logarithmic function<\/strong><br \/>\r\n[latex]\\frac{d}{dx}(\\log_b (g(x)))=\\dfrac{g^{\\prime}(x)}{g(x) \\ln b}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169739296745\" class=\"definition\">\r\n<dt><strong>chain rule<\/strong><\/dt>\r\n<dd id=\"fs-id1169739296750\">the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738191096\" class=\"definition\">\r\n<dt><strong>implicit differentiation<\/strong><\/dt>\r\n<dd id=\"fs-id1169738191101\">is a technique for computing [latex]\\frac{dy}{dx}[\/latex] for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable [latex]y[\/latex] as a function) and solving for [latex]\\frac{dy}{dx}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738074914\" class=\"definition\">\r\n<dt><strong>logarithmic differentiation<\/strong><\/dt>\r\n<dd id=\"fs-id1169738074919\">is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly<\/dd>\r\n<\/dl>\r\n<h2>Study Tips<\/h2>\r\n<p><strong>Derivatives of the Sine and Cosine Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the derivatives of sine and cosine, but understand the proof<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the relationship between sine\/cosine and their derivatives<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Review prerequisite trigonometric identities and limits<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives of Other Trigonometric Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the derivatives of all six trigonometric functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice rewriting complex trigonometric expressions before differentiating<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Review common trigonometric identities, especially Pythagorean identities<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Create a quick reference sheet with trig function values for common angles<\/li>\r\n<\/ul>\r\n<p><strong>Higher-Order Derivatives of Trig Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the 4-step cycle for both sine and cosine<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice quickly calculating remainders for large numbers<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Create visual aids (e.g., circular diagrams) to represent the derivative cycle<\/li>\r\n<\/ul>\r\n<p><strong>Deriving the Chain Rule<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying the inner and outer functions in composite functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the chain rule formula and understand its components.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When applying the chain rule, always work from the outside in.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that derivatives are evaluated at functions, not at other derivatives.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice with a variety of composite functions, including trigonometric, exponential, and root functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the chain rule as a process of \"unwrapping\" nested functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the domain of each function in the composition to ensure the chain rule is applicable.<\/li>\r\n<\/ul>\r\n<p><strong>Combining the Chain Rule With Other Rules<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice recognizing when to use the chain rule in combination with other rules.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When dealing with complex functions, break them down into simpler parts before applying rules.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the order of operations when combining rules.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that these combined rules are just applications of the basic chain rule and other differentiation rules.<\/li>\r\n<\/ul>\r\n<p><strong>Applying the Chain Rule Multiple Times<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Always start differentiating from the outermost function and work inward.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Keep track of where you are in the composition by using parentheses effectively.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that each application of the chain rule introduces a new factor in the derivative.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When dealing with physics problems, clearly identify which function represents position, velocity, or acceleration.<\/li>\r\n<\/ul>\r\n<p><strong>The Chain Rule Using Leibniz\u2019s Notation<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying the intermediate variable ([latex]u[\/latex]) in complex functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the order of multiplication in Leibniz's notation matters: [latex]\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex], not the other way around.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Always express the final answer in terms of the original variable (usually [latex]x[\/latex]).<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives of Various Inverse Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the inverse function theorem allows you to find derivatives without explicitly knowing the inverse function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When dealing with roots, rewrite them as rational exponents before differentiating.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the extended power rule to differentiate expressions with rational exponents.<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives of Inverse Trigonometric Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the derivatives of the six inverse trigonometric functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]\\cos(\\sin^{-1} x) = \\sqrt{1-x^2}[\/latex] for [latex]-1 \\leq x \\leq 1[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to sign changes in derivatives (e.g., between [latex]\\sin^{-1} x[\/latex] and [latex]\\cos^{-1} x[\/latex]).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be prepared to simplify complex expressions resulting from these derivatives.<\/li>\r\n<\/ul>\r\n<p><strong>What is Implicit Differentiation?<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying implicit and explicit functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember to use the chain rule when differentiating terms containing [latex]y[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the order of operations when differentiating complex expressions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be comfortable with algebraic manipulation to isolate [latex]\\frac{dy}{dx}[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Recognize that the final expression for [latex]\\frac{dy}{dx}[\/latex] may contain both [latex]x[\/latex] and [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<p><strong>Finding Tangent Lines Implicitly<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice implicit differentiation on various types of equations (circles, ellipses, etc.).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember to substitute the given point into [latex]\\frac{dy}{dx}[\/latex] to find the slope.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to domain restrictions when working with certain curves.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Review conic sections and their general forms to recognize common implicit equations.<\/li>\r\n<\/ul>\r\n<p><strong>Derivative of the Exponential Function<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Understand why [latex]e[\/latex] is special: it's the only base where [latex]B'(0) = 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice applying the chain rule with [latex]e^x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]e^x[\/latex] is defined for all real numbers<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice finding derivatives of complex expressions involving [latex]e^x[\/latex]<\/li>\r\n<\/ul>\r\n<p><strong>Derivative of the Logarithmic Function<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice using logarithmic properties to simplify expressions before differentiating<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember the relationship between exponential and logarithmic functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the graph of [latex]y = \\ln x[\/latex] and its derivative [latex]y' = \\frac{1}{x}[\/latex]<\/li>\r\n<\/ul>\r\n<p><strong>Logarithmic Differentiation<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying functions that benefit from logarithmic differentiation<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Review and memorize logarithm properties<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the chain rule when differentiating logarithmic expressions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember to multiply by [latex]y[\/latex] when solving for [latex]\\frac{dy}{dx}[\/latex]<\/li>\r\n<\/ul>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Techniques+for+Differentiation.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Derivatives of Trigonometric Functions<\/strong><\/p>\n<ul id=\"fs-id1169739325602\">\n<li>We can find the derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] by using the definition of derivative and the limit formulas found earlier. The results are\n<div id=\"fs-id1169736589261\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx} \\sin x= \\cos x[\/latex]\u00a0 and\u00a0 [latex]\\frac{d}{dx} \\cos x=\u2212\\sin x[\/latex].<\/div>\n<\/li>\n<li>With these two formulas, we can determine the derivatives of all six basic trigonometric functions.<\/li>\n<\/ul>\n<p><strong>The Chain Rule<\/strong><\/p>\n<ul id=\"fs-id1169736594108\">\n<li>The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x))[\/latex],\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/div>\n<ul>\n<li>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">In Leibniz\u2019s notation this rule takes the form<\/div>\n<div id=\"fs-id1169737159954\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.<\/li>\n<li>The chain rule combines with the power rule to form a new rule:\n<div id=\"fs-id1169737470447\" class=\"equation unnumbered\" style=\"text-align: center;\">If [latex]h(x)=(g(x))^n[\/latex], then [latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\n<\/li>\n<li style=\"text-align: left;\">When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x)))[\/latex], then [latex]h^{\\prime}(x)=f^{\\prime}(g(k(x)))g^{\\prime}(k(x))k^{\\prime}(x)[\/latex]<\/li>\n<\/ul>\n<p><strong>Derivatives of Inverse Functions<\/strong><\/p>\n<ul id=\"fs-id1169736611730\">\n<li>The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.<\/li>\n<li>We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.<\/li>\n<\/ul>\n<p><strong>Implicit Differentiation<\/strong><\/p>\n<ul id=\"fs-id1169738240218\">\n<li>We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).<\/li>\n<li>By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.<\/li>\n<\/ul>\n<p><strong>Derivatives of Exponential and Logarithmic Functions<\/strong><\/p>\n<ul id=\"fs-id1169738233616\">\n<li>On the basis of the assumption that the exponential function [latex]y=b^x, \\, b>0[\/latex] is continuous everywhere and differentiable at [latex]0[\/latex], this function is differentiable everywhere and there is a formula for its derivative.<\/li>\n<li>We can use a formula to find the derivative of [latex]y=\\ln x[\/latex], and the relationship [latex]\\log_b x=\\dfrac{\\ln x}{\\ln b}[\/latex] allows us to extend our differentiation formulas to include logarithms with arbitrary bases.<\/li>\n<li>Logarithmic differentiation allows us to differentiate functions of the form [latex]y=g(x)^{f(x)}[\/latex] or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Derivative of sine function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/li>\n<li><strong>Derivative of cosine function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/li>\n<li><strong>Derivative of tangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/li>\n<li><strong>Derivative of cotangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/li>\n<li><strong>Derivative of secant function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/li>\n<li><strong>Derivative of cosecant function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/li>\n<li><strong>The chain rule<\/strong><br \/>\n[latex]\\frac{d}{dx}(f(g(x)))=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>The power rule for functions<\/strong><br \/>\n[latex]\\frac{d}{dx}((g(x)^n)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>Derivative of inverse sine function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\sin^{-1} x)=\\dfrac{1}{\\sqrt{1-x^2}}[\/latex]<\/li>\n<li><strong>Derivative of inverse cosine function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\cos^{-1} x)=\\dfrac{-1}{\\sqrt{1-x^2}}[\/latex]<\/li>\n<li><strong>Derivative of inverse tangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\tan^{-1} x)=\\dfrac{1}{1+x^2}[\/latex]<\/li>\n<li><strong>Derivative of inverse cotangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\cot^{-1} x)=\\dfrac{-1}{1+x^2}[\/latex]<\/li>\n<li><strong>Derivative of inverse secant function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\sec^{-1} x)=\\dfrac{1}{|x|\\sqrt{x^2-1}}[\/latex]<\/li>\n<li><strong>Derivative of inverse cosecant function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\csc^{-1} x)=\\dfrac{-1}{|x|\\sqrt{x^2-1}}[\/latex]<\/li>\n<li><strong>Inverse function theorem<\/strong><br \/>\n[latex](f^{-1})^{\\prime}(x)=\\dfrac{1}{f^{\\prime}(f^{-1}(x))}[\/latex] whenever [latex]f^{\\prime}(f^{-1}(x))\\ne 0[\/latex] and [latex]f(x)[\/latex] is differentiable.<\/li>\n<li><strong>Power rule with rational exponents<\/strong><br \/>\n[latex]\\frac{d}{dx}(x^{m\/n})=\\frac{m}{n}x^{(m\/n)-1}[\/latex].<\/li>\n<li><strong>Derivative of the natural exponential function<\/strong><br \/>\n[latex]\\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>Derivative of the natural logarithmic function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\ln (g(x)))=\\dfrac{1}{g(x)} g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>Derivative of the general exponential function<\/strong><br \/>\n[latex]\\frac{d}{dx}(b^{g(x)})=b^{g(x)} g^{\\prime}(x) \\ln b[\/latex]<\/li>\n<li><strong>Derivative of the general logarithmic function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\log_b (g(x)))=\\dfrac{g^{\\prime}(x)}{g(x) \\ln b}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169739296745\" class=\"definition\">\n<dt><strong>chain rule<\/strong><\/dt>\n<dd id=\"fs-id1169739296750\">the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738191096\" class=\"definition\">\n<dt><strong>implicit differentiation<\/strong><\/dt>\n<dd id=\"fs-id1169738191101\">is a technique for computing [latex]\\frac{dy}{dx}[\/latex] for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable [latex]y[\/latex] as a function) and solving for [latex]\\frac{dy}{dx}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738074914\" class=\"definition\">\n<dt><strong>logarithmic differentiation<\/strong><\/dt>\n<dd id=\"fs-id1169738074919\">is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>Derivatives of the Sine and Cosine Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the derivatives of sine and cosine, but understand the proof<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the relationship between sine\/cosine and their derivatives<\/li>\n<li class=\"whitespace-normal break-words\">Review prerequisite trigonometric identities and limits<\/li>\n<\/ul>\n<p><strong>Derivatives of Other Trigonometric Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the derivatives of all six trigonometric functions<\/li>\n<li class=\"whitespace-normal break-words\">Practice rewriting complex trigonometric expressions before differentiating<\/li>\n<li class=\"whitespace-normal break-words\">Review common trigonometric identities, especially Pythagorean identities<\/li>\n<li class=\"whitespace-normal break-words\">Create a quick reference sheet with trig function values for common angles<\/li>\n<\/ul>\n<p><strong>Higher-Order Derivatives of Trig Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the 4-step cycle for both sine and cosine<\/li>\n<li class=\"whitespace-normal break-words\">Practice quickly calculating remainders for large numbers<\/li>\n<li class=\"whitespace-normal break-words\">Create visual aids (e.g., circular diagrams) to represent the derivative cycle<\/li>\n<\/ul>\n<p><strong>Deriving the Chain Rule<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying the inner and outer functions in composite functions.<\/li>\n<li class=\"whitespace-normal break-words\">Memorize the chain rule formula and understand its components.<\/li>\n<li class=\"whitespace-normal break-words\">When applying the chain rule, always work from the outside in.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that derivatives are evaluated at functions, not at other derivatives.<\/li>\n<li class=\"whitespace-normal break-words\">Practice with a variety of composite functions, including trigonometric, exponential, and root functions.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the chain rule as a process of &#8220;unwrapping&#8221; nested functions.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the domain of each function in the composition to ensure the chain rule is applicable.<\/li>\n<\/ul>\n<p><strong>Combining the Chain Rule With Other Rules<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice recognizing when to use the chain rule in combination with other rules.<\/li>\n<li class=\"whitespace-normal break-words\">When dealing with complex functions, break them down into simpler parts before applying rules.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the order of operations when combining rules.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that these combined rules are just applications of the basic chain rule and other differentiation rules.<\/li>\n<\/ul>\n<p><strong>Applying the Chain Rule Multiple Times<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Always start differentiating from the outermost function and work inward.<\/li>\n<li class=\"whitespace-normal break-words\">Keep track of where you are in the composition by using parentheses effectively.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that each application of the chain rule introduces a new factor in the derivative.<\/li>\n<li class=\"whitespace-normal break-words\">When dealing with physics problems, clearly identify which function represents position, velocity, or acceleration.<\/li>\n<\/ul>\n<p><strong>The Chain Rule Using Leibniz\u2019s Notation<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying the intermediate variable ([latex]u[\/latex]) in complex functions.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the order of multiplication in Leibniz&#8217;s notation matters: [latex]\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex], not the other way around.<\/li>\n<li class=\"whitespace-normal break-words\">Always express the final answer in terms of the original variable (usually [latex]x[\/latex]).<\/li>\n<\/ul>\n<p><strong>Derivatives of Various Inverse Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Remember that the inverse function theorem allows you to find derivatives without explicitly knowing the inverse function.<\/li>\n<li class=\"whitespace-normal break-words\">When dealing with roots, rewrite them as rational exponents before differentiating.<\/li>\n<li class=\"whitespace-normal break-words\">Use the extended power rule to differentiate expressions with rational exponents.<\/li>\n<\/ul>\n<p><strong>Derivatives of Inverse Trigonometric Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the derivatives of the six inverse trigonometric functions.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]\\cos(\\sin^{-1} x) = \\sqrt{1-x^2}[\/latex] for [latex]-1 \\leq x \\leq 1[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to sign changes in derivatives (e.g., between [latex]\\sin^{-1} x[\/latex] and [latex]\\cos^{-1} x[\/latex]).<\/li>\n<li class=\"whitespace-normal break-words\">Be prepared to simplify complex expressions resulting from these derivatives.<\/li>\n<\/ul>\n<p><strong>What is Implicit Differentiation?<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying implicit and explicit functions.<\/li>\n<li class=\"whitespace-normal break-words\">Remember to use the chain rule when differentiating terms containing [latex]y[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the order of operations when differentiating complex expressions.<\/li>\n<li class=\"whitespace-normal break-words\">Be comfortable with algebraic manipulation to isolate [latex]\\frac{dy}{dx}[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Recognize that the final expression for [latex]\\frac{dy}{dx}[\/latex] may contain both [latex]x[\/latex] and [latex]y[\/latex].<\/li>\n<\/ul>\n<p><strong>Finding Tangent Lines Implicitly<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice implicit differentiation on various types of equations (circles, ellipses, etc.).<\/li>\n<li class=\"whitespace-normal break-words\">Remember to substitute the given point into [latex]\\frac{dy}{dx}[\/latex] to find the slope.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to domain restrictions when working with certain curves.<\/li>\n<li class=\"whitespace-normal break-words\">Review conic sections and their general forms to recognize common implicit equations.<\/li>\n<\/ul>\n<p><strong>Derivative of the Exponential Function<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand why [latex]e[\/latex] is special: it&#8217;s the only base where [latex]B'(0) = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Practice applying the chain rule with [latex]e^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]e^x[\/latex] is defined for all real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Practice finding derivatives of complex expressions involving [latex]e^x[\/latex]<\/li>\n<\/ul>\n<p><strong>Derivative of the Logarithmic Function<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice using logarithmic properties to simplify expressions before differentiating<\/li>\n<li class=\"whitespace-normal break-words\">Remember the relationship between exponential and logarithmic functions<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the graph of [latex]y = \\ln x[\/latex] and its derivative [latex]y' = \\frac{1}{x}[\/latex]<\/li>\n<\/ul>\n<p><strong>Logarithmic Differentiation<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying functions that benefit from logarithmic differentiation<\/li>\n<li class=\"whitespace-normal break-words\">Review and memorize logarithm properties<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the chain rule when differentiating logarithmic expressions<\/li>\n<li class=\"whitespace-normal break-words\">Remember to multiply by [latex]y[\/latex] when solving for [latex]\\frac{dy}{dx}[\/latex]<\/li>\n<\/ul>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":560,"module-header":"cheat_sheet","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3595"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3595\/revisions"}],"predecessor-version":[{"id":3956,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3595\/revisions\/3956"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3595\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3595"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3595"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3595"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3595"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}