Chain Rule Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Chain Rule.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The chain rule differentiates a composite $f(g(x))$ as $f'(g(x))$ times $g'(x)$ , peeling layers from outside in.

Common stuck point: The procedure for chain rule is the easy part; the trap is forgetting the inner-derivative factor $g'(x)$ . Asking "Is one function plugged into another, so I must multiply the outer derivative by the inner derivative?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is one function plugged into another, so I must multiply the outer derivative by the inner derivative?

Worked Examples

Example 1

easy

Find the derivative of

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

Answer

f'(x) = 12(3x + 1)^3

First step

Identify the outer function

u^4

and the inner function

u = 3x + 1

Full solution

2
Apply the chain rule: $\frac{d}{dx}[u^4] = 4u^3 \cdot \frac{du}{dx}$ .
3
The derivative of the inner function: $\frac{du}{dx} = 3$ .
4
Combine: $f'(x) = 4(3x+1)^3 \cdot 3 = 12(3x+1)^3$ .

The chain rule says: differentiate the outer function, keep the inner function, then multiply by the derivative of the inner function. Think of it as peeling layers.

Example 2

medium

Find the derivative of

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

Example 3

hard

Find the derivative of

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

Example 4

medium

Differentiate

f(x) = \sin^3 x

Example 5

medium

Differentiate

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

Example 6

medium

Differentiate

f(x) = e^{x^2 + 3x}

Example 7

medium

Differentiate

f(x) = \sin(\cos x)

Example 8

hard

Differentiate

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

Example 9

hard

y = \tan^{-1}(2x)

, find

\tfrac{dy}{dx}

Example 10

hard

A spherical balloon's radius grows at

2

cm/s. How fast is its volume changing when

r = 5

cm?

Example 11

challenge

Let

f

be differentiable with

f(1) = 2

and

f'(1) = 3

. If

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

, find

g'(1)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the derivative of

f(x) = (5x - 2)^3

Example 2

medium

Find the derivative of

f(x) = \sqrt{4x + 3}

Example 3

easy

Differentiate

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

Example 4

easy

Differentiate

f(x) = \sin(2x)

Example 5

easy

Differentiate

f(x) = e^{3x}

Example 6

easy

Differentiate

f(x) = (2x-5)^4

Example 7

easy

Differentiate

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

Example 8

easy

Differentiate

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

Example 9

easy

Differentiate

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

Example 10

easy

Differentiate

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

Example 11

medium

Differentiate

f(x) = \sin^2 x

Example 12

medium

Differentiate

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

(two nested layers).

Example 13

medium

Differentiate

f(x) = (3x^2 + 2x)^5

Example 14

medium

Differentiate

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

Example 15

medium

Differentiate

f(x) = \tan(3x)

Example 16

medium

Differentiate

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

Example 17

medium

Differentiate

f(x) = \sqrt{\sin x}

Example 18

challenge

Differentiate

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

(three layers).

Example 19

challenge

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

with

g(2) = 5

g'(2) = 3

f'(5) = 4

, find

h'(2)

Example 20

challenge

Differentiate

f(x) = \frac{1}{(x^2+1)^3}

Example 21

medium

Differentiate

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

(chain and product rules).

Example 22

medium

Differentiate

f(x) = \sin(\cos x)

Example 23

easy

Differentiate

f(x) = (3x + 7)^2

Example 24

easy

Differentiate

f(x) = e^{5x}

Example 25

easy

Differentiate

f(x) = (1 - x)^7

Example 26

easy

Differentiate

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

Example 27

medium

Differentiate

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

Example 28

medium

Differentiate

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

Example 29

medium

Differentiate

f(x) = \sqrt{\sin x}

Example 30

medium

Find

\tfrac{dy}{dx}

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

x = 1

Example 31

hard

Differentiate

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

Example 32

hard

Differentiate

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

Example 33

hard

Find

\tfrac{dy}{dx}

for

y = (\ln(3x^2 + 1))^5

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Related Concepts

Derivative Function Composition

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativecomposition