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Chain Rule: Problems and Solutions

Chain Rule: Problems and Solutions

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.

Want to skip the Summary? Jump down to problems and their solutions.

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.

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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 $f(x) = \left(3x^2 – 4x + 5\right)^8.$
Click to View Calculus Solution
Chain Rule Problem #2
Differentiate $f(x) = \tan^3 x.$
Hint: Recall $\tan^3 x = \big[\tan x\big]^3.$ Also recall that $\dfrac{d}{dx}\tan x = \sec^2 x.$
Click to View Calculus Solution
Chain Rule Problem #3
Differentiate $f(x) = (\cos x – \sin x)^{-2}.$
Click to View Calculus Solution
Chain Rule Problem #4
Differentiate $f(x) = \left(x^5 + e^x\right)^{99}.$
Click to View Calculus Solution
Chain Rule Problem #5
Differentiate $f(x) = \sqrt{x^2+1}.$

Click to View Calculus Solution
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 $f(x) = e^{\sin x}.$
Click to View Calculus Solution
Chain Rule Problem #7
Differentiate $f(x) = e^{\left(x^7 – 4x^3 + x \right)}.$
Click to View Calculus Solution
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 $f(x) = \sin(2x).$
Click to View Calculus Solution
Chain Rule Problem #9
Differentiate $f(x) = \tan(e^x).$
Click to View Calculus Solution
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
This problem combines the Product Rule with the Chain Rule.
Differentiate $f(x) = \left(x^2 + 1 \right)^7 (3x – 7)^4.$
Click to View Calculus Solution
Chain Rule Problem #11
This problem requires using the Chain Rule twice.
Differentiate $f(x) = \cos(\tan(3x)).$
Click to View Calculus Solution
Chain Rule Problem #12
This problem requires using the Chain Rule three times.
Differentiate $f(x) = \left(1 + \sin^9(2x + 3) \right)^2.$
Hint: Recall that $\sin^9(\cdots) = \big[\sin(\cdots) \big]^9.$
Click to View Calculus Solution
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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