Practice Chain Rule in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The derivative of a composite function f(g(x))f(g(x)) equals f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.

Derivative of outside times derivative of inside. Unpack layers.

Showing a random 20 of 50 problems.

Example 1

medium
Differentiate f(x)=sin⁡(cos⁡x)f(x) = \sin(\cos x).

Example 2

easy
Find the derivative of f(x)=(3x+1)4f(x) = (3x + 1)^4.

Example 3

easy
Differentiate f(x)=cos⁡(x2+1)f(x) = \cos(x^2 + 1).

Example 4

medium
Differentiate f(x)=sin⁡(e3x)f(x) = \sin(e^{3x}) (two nested layers).

Example 5

challenge
If h(x)=f(g(x))h(x) = f(g(x)) with g(2)=5g(2) = 5, g′(2)=3g'(2) = 3, f′(5)=4f'(5) = 4, find h′(2)h'(2).

Example 6

easy
Differentiate f(x)=(x2+4)1/2f(x) = (x^2 + 4)^{1/2}.

Example 7

medium
Differentiate f(x)=tan⁡(3x2)f(x) = \tan(3x^2).

Example 8

easy
Differentiate f(x)=e3xf(x) = e^{3x}.

Example 9

medium
If f(x)=cos⁡(ln⁡x)f(x) = \cos(\ln x), find f′(x)f'(x).

Example 10

challenge
Differentiate f(x)=esin⁡(x2)f(x) = e^{\sin(x^2)} (three layers).

Example 11

medium
Differentiate f(x)=cos⁡3(x)f(x) = \cos^3(x).

Example 12

medium
Differentiate f(x)=sin⁡3xf(x) = \sin^3 x.

Example 13

hard
Differentiate f(x)=(2x+1)3(x2−1)2f(x) = (2x + 1)^3 (x^2 - 1)^2.

Example 14

medium
Find the derivative of f(x)=(x2+1)5f(x) = (x^2 + 1)^5.

Example 15

easy
Differentiate f(x)=(x+1)3f(x) = (x+1)^3.

Example 16

medium
Find dydx\tfrac{dy}{dx} if y=(x2+1)10y = (x^2 + 1)^{10} at x=1x = 1.

Example 17

medium
Differentiate f(x)=esin⁡xf(x) = e^{\sin x}.

Example 18

hard
Differentiate f(x)=esin⁡(2x)f(x) = e^{\sin(2x)}.

Example 19

medium
Differentiate f(x)=sin⁡2xf(x) = \sin^2 x.

Example 20

hard
Differentiate f(x)=ln⁡(x+x2+1)f(x) = \ln(x + \sqrt{x^2 + 1}).
← Back to Chain Rule Worked Examples Formula Explained