Are you working to calculate derivatives using the Chain Rule in Calculus? Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself.

Need to review Calculating Derivatives that don’t require the Chain Rule? That material is here.

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CALCULUS SUMMARY: Chain Rule

You can always access our Handy Table of Derivatives and Differentiation Rules via the Key Formulas menu item at the top of every page.

Chain rule & Power rule

\begin{align*}

\text{If} && f(x) &= (\text{stuff})^n, \\[8px]

\text{then} &&\dfrac{df}{dx} &= n(\text{that stuff})^{n-1} \cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\left(u^n \right) = n u^{n-1} \cdot \dfrac{du}{dx}$$

The following five problems illustrate.

Chain Rule Problem #1

Differentiate $f(x) = \left(3x^2 – 4x + 5\right)^8.$

Chain Rule Problem #2

Differentiate $f(x) = \tan^3 x.$

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Chain Rule Problem #3

Differentiate $f(x) = (\cos x – \sin x)^{-2}.$

Chain Rule Problem #4

Differentiate $f(x) = \left(x^5 + e^x\right)^{99}.$

Chain Rule Problem #5

Differentiate $f(x) = \sqrt{x^2+1}.$

Chain rule & Exponentials

\begin{align*}

\text{If} && f(x) &= e^{\text{(stuff)}}, \\[8px]

\text{then} &&\dfrac{df}{dx} &= e^{\text{(that stuff)}}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}e^u = e^u \cdot \dfrac{du}{dx}$$

The next two problems illustrate.

Chain Rule Problem #6

Differentiate $f(x) = e^{\sin x}.$

Chain Rule Problem #7

Differentiate $f(x) = e^{\left(x^7 – 4x^3 + x \right)}.$

Chain rule & Trig Functions

\begin{align*}

\text{If} && f(x) &= \sin\text{(stuff)}, \\[8px]

\text{then} &&\dfrac{df}{dx} &= \cos\text{(that stuff)}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\sin u = \cos u \cdot \dfrac{du}{dx}$$

$$ — $$

\begin{align*}

\text{If} && f(x) &= \cos\text{(stuff)}, \\[8px]

\text{then} &&\dfrac{df}{dx} &= -\sin\text{(that stuff)}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\cos u = -\sin u \cdot \dfrac{du}{dx}$$

$$ — $$

\begin{align*}

\text{If} && f(x) &= \tan\text{(stuff)}, \\[8px]

\text{then} &&\dfrac{df}{dx} &= \sec^2\text{(that stuff)}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\tan u = \sec^2 u \cdot \dfrac{du}{dx}$$

The next two problems illustrate.

Chain Rule Problem #8

Differentiate $f(x) = \sin(2x).$

To access more problems and solutions, including AP-style multiple choice questions, log in for free with your Google, Apple or Facebook account, or create a dedicated Matheno in 60 seconds. You’ll also then be able to mark the problems *you* want to be sure to review before your exams. Simply use the log-in area at the bottom of this screen.

Chain Rule Problem #9

Differentiate $f(x) = \tan(e^x).$

Harder Chain Rule Problems

The problems below combine the Product rule and the Chain rule, or require using the Chain rule multiple times.

Chain Rule Problem #10

Differentiate $f(x) = \left(x^2 + 1 \right)^7 (3x – 7)^4.$

Chain Rule Problem #11

Differentiate $f(x) = \cos(\tan(3x)).$

Chain Rule Problem #12

Differentiate $f(x) = \left(1 + \sin^9(2x + 3) \right)^2.$

Need to use the derivative to find the equation of a tangent line (or the equation of a normal line)? We have a separate page on that topic here.

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