Practice Chain Rule in Math

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

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Quick Recap

The derivative of a composite function $f(g(x))$ equals $f'(g(x)) \cdot g'(x)$ : 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)

Example 2

easy

Find the derivative of

f(x) = (3x + 1)^4

Example 3

easy

Differentiate

f(x) = \cos(x^2 + 1)

Example 4

medium

Differentiate

f(x) = \sin(e^{3x})

(two nested layers).

Example 5

challenge

h(x) = f(g(x))

with

g(2) = 5

g'(2) = 3

f'(5) = 4

, find

h'(2)

Example 6

easy

Differentiate

f(x) = (x^2 + 4)^{1/2}

Example 7

medium

Differentiate

f(x) = \tan(3x^2)

Example 8

easy

Differentiate

f(x) = e^{3x}

Example 9

medium

f(x) = \cos(\ln x)

, find

f'(x)

Example 10

challenge

Differentiate

f(x) = e^{\sin(x^2)}

(three layers).

Example 11

medium

Differentiate

f(x) = \cos^3(x)

Example 12

medium

Differentiate

f(x) = \sin^3 x

Example 13

hard

Differentiate

f(x) = (2x + 1)^3 (x^2 - 1)^2

Example 14

medium

Find the derivative of

f(x) = (x^2 + 1)^5

Example 15

easy

Differentiate

f(x) = (x+1)^3

Example 16

medium

Find

\tfrac{dy}{dx}

y = (x^2 + 1)^{10}

x = 1

Example 17

medium

Differentiate

f(x) = e^{\sin x}

Example 18

hard

Differentiate

f(x) = e^{\sin(2x)}

Example 19

medium

Differentiate

f(x) = \sin^2 x

Example 20

hard

Differentiate

f(x) = \ln(x + \sqrt{x^2 + 1})

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Related Concepts

Derivative Function Composition