The chain rule is used to find derivatives of composite functions, where one function is inside another function. This page focuses on recognizing the outside and inside functions, differentiating the outside first, and then multiplying by the derivative of the inside. The practice problems include powers, radicals, trigonometric functions, exponential functions, logarithms, and table-based chain rule questions.

Notes

Practice Problems

Find the derivative of each

\(\small{\textbf{1)}}\) Find \(f'(x)\) where \(f(x)=(3x+5)^4\)

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\(\small{\textbf{2)}}\) Find \(f'(x)\) where \(f(x)=\sqrt{2x-1}\)

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\(\small{\textbf{3)}}\) Find \(f'(x)\) where \(f(x)=\displaystyle\frac{1}{x^3+3x-2}\)

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\(\small{\textbf{4)}}\) Find \(f'(x)\) where \(f(x)=x^2(2x-1)^3\)

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\(\small{\textbf{5)}}\) Find \(f'(x)\) where \(f(x)=\sqrt[3]{x^4-2x+1}\)

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\(\small{\textbf{6)}}\) Find \(f'(x)\) where \(f(x)=(x^3-2)^4\)

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\(\small{\textbf{7)}}\) Find \(f'(x)\) where \(f(x) = (3x^2 + 5)^6\)

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\(\small{\textbf{8)}}\) Find \(f'(x)\) where \(f(x) = \displaystyle\frac{1}{\left( x^2 + 2x – 5 \right)^ 3}\)

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\(\small{\textbf{9)}}\) Find \(f'(x)\) where \(f(x) = \sqrt[5]{{1 – x}}\)

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\(\small{\textbf{10)}}\) Find \(f'(x)\) where \(f(x) = \csc \left( {2x} \right)\)

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\(\small{\textbf{11)}}\) Find \(f'(x)\) where \(f(x) = 3\cos \left( {\tan \left( 4x \right)} \right)\)

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\(\small{\textbf{12)}}\) Find \(f'(x)\) where \(f(x) = \tan \left( {15x + 14} \right)\)

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\(\small{\textbf{13)}}\) Find \(f'(x)\) where \(f(x) = e^{(x^2 + 2)}\)

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\(\small{\textbf{14)}}\) Find \(f'(x)\) where \(f(x) = e^{1 – \sin \left( x \right)}\)

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\(\small{\textbf{15)}}\) Find \(f'(x)\) where \(f(x) = {3^{1 + 4x}}\)

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\(\small{\textbf{16)}}\) Find \(f'(x)\) where \(f(x) = {\tan ^{ – 1}}\left( {5x + 2} \right)\)

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\(\small{\textbf{17)}}\) Find \(f'(x)\) where \(f(x) = \ln \left({2x^3 +3x +6} \right)\)

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\(\small{\textbf{18)}}\) Find \(f'(x)\) where \(f(x) = \ln \left( {\sin \left( x \right) + \cos \left( x \right)} \right)\)

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\(\small{\textbf{19)}}\) Find \(f'(x)\) where \(f(x) = \ln \left( {\tan \left( x \right) + \cot \left( x \right)} \right)\)

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\(\small{\textbf{20)}}\) Find \(f'(x)\) where \(f(x) = {\sin ^2}\left( x \right) + \sin \left( {{x^2}} \right)\)

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\(\small{\textbf{21)}}\) Find \(f'(x)\) where \(f(x) = {e^{ – x}}+e^x\)

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\(\small{\textbf{22)}}\) Find \(f'(x)\) where \(f(x) = 2x^3 + \left(4x^5 – 6x \right)^7\)

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\(\small{\textbf{23)}}\) Find \(f'(x)\) where \(f(x) = \ln \left( {\sin \left( x \right)} \right) – \sin \left( \ln x \right) \)

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\(\small{\textbf{24)}}\) Find \(f'(x)\) where \(f(x) = x^2\ln(x^2)\)

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\(\small{\textbf{25)}}\) Find \(f'(x)\) where \(f(x) = \sin \left( {2x} \right)\cos \left( {2x} \right)\)

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\(\small{\textbf{26)}}\) Find \(f'(x)\) where \( f(x) = \sin \left( {\cos \left( {2x} \right)} \right)\)

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\(\small{\textbf{27)}}\) Find \(f'(x)\) where \(f(x) = \displaystyle\frac{\sin {2x} }{\cos{2x}}\)

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\(\small{\textbf{28)}}\) Find \(f'(x)\) where \(f(x) = \sin \left( e^x \right)\)

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\(\small{\textbf{29)}}\) Find \(f'(x)\) where \(f(x) = \sqrt {\tan \left( {2x} \right)}\)

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\(\small{\textbf{30)}}\) Find \(f'(x)\) where \(f(x) = {\tan ^3}\left( {{x^3}} \right)\)

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\(\small{\textbf{31)}}\) Find \(f'(x)\) where \(f(x)=\ln⁡(\ln⁡ x)\)

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\(\small{\textbf{32)}}\) Find \(f'(x)\) where \(f(x)=\displaystyle 3^{\sin^4(x^3)}\)

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\(\small{\textbf{33)}}\) Find \(f'(x)\) where \(f(x)=\displaystyle \sec{\left(\sin(x)\right)}\)

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\(\small{\textbf{34)}}\) Find \(f'(x)\) where \(f(x)=\displaystyle \ln{\sqrt{x}}\)

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\(\small{\textbf{35)}}\) Find \(f'(3)\) where \(f(x)=g(h(x))\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)
\(g(x)\)	\(9\)	\(4\)	\(6\)	\(2\)
\(g'(x)\)	\(8\)	\(3\)	\(5\)	\(2\)
\(h(x)\)	\(1\)	\(3\)	\(4\)	\(2\)
\(h'(x)\)	\(4\)	\(-2\)	\(7\)	\(3\)

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\(\small{\textbf{36)}}\) Find \(f'(1)\) where \(f(x)=g(h(x))\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)
\(g(x)\)	\(9\)	\(4\)	\(6\)	\(2\)
\(g'(x)\)	\(8\)	\(3\)	\(5\)	\(2\)
\(h(x)\)	\(1\)	\(3\)	\(4\)	\(2\)
\(h'(x)\)	\(4\)	\(-2\)	\(7\)	\(3\)

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\(\small{\textbf{37)}}\) Find \(f'(2)\) where \(f(x)=g(h(x))\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)
\(g(x)\)	\(9\)	\(4\)	\(6\)	\(2\)
\(g'(x)\)	\(8\)	\(3\)	\(5\)	\(2\)
\(h(x)\)	\(1\)	\(3\)	\(4\)	\(2\)
\(h'(x)\)	\(4\)	\(-2\)	\(7\)	\(3\)

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\(\small{\textbf{38)}}\) Find \(f'(4)\) where \(f(x)=g(h(x))\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)
\(g(x)\)	\(9\)	\(4\)	\(6\)	\(2\)
\(g'(x)\)	\(8\)	\(3\)	\(5\)	\(2\)
\(h(x)\)	\(1\)	\(3\)	\(4\)	\(2\)
\(h'(x)\)	\(4\)	\(-2\)	\(7\)	\(3\)

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See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)