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The Chain Rule

Deciding when a composition genuinely needs the chain rule, confirming equivalent forms give the same derivative, and simplifying with log or trig identities before differentiating.

Calculus Lab Manual, The Chain Rule

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31 exercises

Exercise 1

The functions f(x)=e3x\fe{f}{x}=e^{3x}, g(x)=(ex)3\fe{g}{x}=\left(e^x\right)^3, and h(x)=(e3)x\fe{h}{x}=\left(e^3\right)^x are equivalent. Find the first derivative formulas for each of the functions (without altering their given forms) and then explicitly establish that the derivative formulas are the same.

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Exercise 2

Consider the function k(θ)=sin⁡−1(sin⁡(θ))\fe{k}{\theta}=\fe{\sin^{-1}}{\fe{\sin}{\theta}}.

Over the interval [−π2,π2]\cinterval{-\frac{\pi}{2}}{\frac{\pi}{2}}, k(θ)=θ\fe{k}{\theta}=\theta. What does this tell you about the formula for k′(θ)\fe{\fd{k}}{\theta} over (−π2,π2)\ointerval{-\frac{\pi}{2}}{\frac{\pi}{2}}?

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Exercise 3

Consider the function k(θ)=sin⁡−1(sin⁡(θ))\fe{k}{\theta}=\fe{\sin^{-1}}{\fe{\sin}{\theta}}.

Use the chain rule and an appropriate trigonometric identity to verify your answer to this exercise. Please note that cos⁡(θ)>0\fe{\cos}{\theta}\gt0 over (−π2,π2)\ointerval{-\frac{\pi}{2}}{\frac{\pi}{2}}.

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Exercise 4

Consider the function k(θ)=sin⁡−1(sin⁡(θ))\fe{k}{\theta}=\fe{\sin^{-1}}{\fe{\sin}{\theta}}.

What is the constant value of k′(θ)\fe{\fd{k}}{\theta} over (π2,3π2)\ointerval{\frac{\pi}{2}}{\frac{3\pi}{2}}?

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Exercise 5

Consider the function k(θ)=sin⁡−1(sin⁡(θ))\fe{k}{\theta}=\fe{\sin^{-1}}{\fe{\sin}{\theta}}.

What is the value of k′(π2)\fe{\fd{k}}{\frac{\pi}{2}}?

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Exercise 10

Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to xx of each of the following functions.

f(x)=ln⁡(x+1)\fe{f}{x}=\fe{\ln}{x+1}

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Exercise 11

Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to xx of each of the following functions.

f(x)=2x5\fe{f}{x}=\dfrac{2}{x^5}

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Exercise 12

Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to xx of each of the following functions.

f(x)=cos⁡(π)\fe{f}{x}=\fe{\cos}{\pi}

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Exercise 13

Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to xx of each of the following functions.

f(x)=cos⁡(x)\fe{f}{x}=\fe{\cos}{x}

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Exercise 14

Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to xx of each of the following functions.

f(x)=cos⁡(πx)\fe{f}{x}=\fe{\cos}{\pi x}

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Exercise 15

Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to xx of each of the following functions.

f(x)=cos⁡(πx)π\fe{f}{x}=\dfrac{\fe{\cos}{\pi x}}{\pi}

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Exercise 16

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=tan⁡−1(x)\fe{f}{x}=\fe{\tan^{-1}}{\sqrt{x}}

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Exercise 17

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=eesin⁡(x)\fe{f}{x}=e^{e^{\fe{\sin}{x}}}

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Exercise 18

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=sin⁡−1(cos⁡(x))\fe{f}{x}=\fe{\sin^{-1}}{\fe{\cos}{x}}

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Exercise 19

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=tan⁡(xsec⁡(x))\fe{f}{x}=\fe{\tan}{x\fe{\sec}{x}}

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Exercise 20

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=tan⁡(x)sec⁡(sec⁡(x))\fe{f}{x}=\fe{\tan}{x}\fe{\sec}{\fe{\sec}{x}}

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Exercise 21

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=(sin⁡(x2))23\fe{f}{x}=\sqrt[3]{\left(\fe{\sin}{x^2}\right)^2}

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Exercise 22

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=4xsin⁡2(x)\fe{f}{x}=4x\fe{\sin^2}{x}

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Exercise 23

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=ln⁡(xln⁡(x))\fe{f}{x}=\fe{\ln}{x\fe{\ln}{x}}

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Exercise 24

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=ln⁡(5xex)\fe{f}{x}=\fe{\ln}{\dfrac{5}{xe^x}}

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Exercise 25

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=2ln⁡(xtan⁡2(x)3)\fe{f}{x}=2\fe{\ln}{\sqrt[3]{x\fe{\tan^2}{x}}}

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Exercise 26

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=ln⁡(ex+2x+2)\fe{f}{x}=\fe{\ln}{\dfrac{e^{x+2}}{\sqrt{x+2}}}

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Exercise 27

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=ln⁡(xe+e)\fe{f}{x}=\fe{\ln}{x^e+e}

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Exercise 28

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=sec⁡4(ex)\fe{f}{x}=\fe{\sec^4}{e^x}

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Exercise 29

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=sec⁡−1(ex)\fe{f}{x}=\fe{\sec^{-1}}{e^x}

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Exercise 32

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=tan⁡−1(2x)2\fe{f}{x}=\dfrac{\fe{\tan^{-1}}{2x}}{2}

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Exercise 33

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=x3sin⁡(x3)\fe{f}{x}=x^3\fe{\sin}{\dfrac{x}{3}}

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Exercise 35

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=exexx\fe{f}{x}=\dfrac{e^{xe^x}}{x}

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Exercise 36

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=xexe2\fe{f}{x}=xe^{xe^2}

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Exercise 38

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=4xsin⁡(x)cos⁡(x2)\fe{f}{x}=4x\fe{\sin}{x}\fe{\cos}{x^2}

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Exercise 39

Find the first derivative with respect to xx of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.

f(x)=sin⁡(xcos⁡2(x))\fe{f}{x}=\fe{\sin}{x\fe{\cos^2}{x}}

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Adapted from Calculus Lab Manual © 2012–2016 Portland Community College. Licensed under CC BY-SA 4.0. Source: Calculus Lab Manual. Imported at revision 3060f8e.