Calculus · Step-by-step guide

Chain Rule Examples

Differentiate composite functions with the chain rule, from powers to trig and exponential expressions.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

For a composition f(g(x))\displaystyle f(g(x)), differentiate the outer function while leaving the inner function unchanged, then multiply by the inner derivative: (f(g(x)))′=f′(g(x))g′(x)\displaystyle (f(g(x)))'=f'(g(x))g'(x).

ddxf(g(x))=f′(g(x))g′(x)\displaystyle \frac d{dx}f(g(x))=f'(g(x))g'(x)
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Identify layers

Ask what operation happens last; that is the outer function. The expression it acts on is the inner function.

2

Differentiate outside-in

Differentiate the outer layer, copy the inner expression, then multiply by the inner derivative.

3

Repeat for nested layers

A function inside a function inside another function may require the chain rule more than once.

02

See the structure

A picture makes the relationships easier to remember.

tangent line
The derivative at the marked point is the slope of the highlighted tangent line.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Power composition

Differentiate (3x2−1)5\displaystyle (3x^2-1)^5.

  1. Outer derivative: 5(3x2−1)4\displaystyle 5(3x^2-1)^4.
  2. Inner derivative: 6x\displaystyle 6x.
  3. Multiply.
Answer30x(3x2−1)4\displaystyle 30x(3x^2-1)^4
Example 2Trig composition

Differentiate sin⁡(4x)\displaystyle \sin(4x).

  1. Outer derivative is cosine.
  2. Keep the inner 4x.
  3. Multiply by derivative 4.
Answer4cos⁡(4x)\displaystyle 4\cos(4x)
Example 3Exponential composition

Differentiate ex3+2x\displaystyle e^{x^3+2x}.

  1. The outer derivative remains exponential.
  2. Differentiate exponent: 3x2+2\displaystyle 3x^2+2.
  3. Multiply the factors.
Answer(3x2+2)ex3+2x\displaystyle (3x^2+2)e^{x^3+2x}
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Differentiating the inside but forgetting to multiply it.

  • Changing the inner expression while differentiating the outer layer.

  • Stopping after only one layer in a multi-layer composition.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Chain Rule

Practice recognizing and differentiating nested functions.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Differentiate (3x+1)5\displaystyle (3x+1)^5.

Show hint

Outer power, then inner derivative.

Show worked solution
  1. Outer derivative: 5(3x+1)4\displaystyle 5(3x+1)^4.
  2. Multiply by d(3x+1)/dx=3\displaystyle d(3x+1)/dx=3.

Answer: 15(3x+1)4\displaystyle 15(3x+1)^4

2

Differentiate sin⁡(x2)\displaystyle \sin(x^2).

Show hint

Differentiate sine, keep the inside.

Show worked solution
  1. Outer derivative: cos⁡(x2)\displaystyle \cos(x^2).
  2. Multiply by inner derivative 2x\displaystyle 2x.

Answer: 2xcos⁡(x2)\displaystyle 2x\cos(x^2)

3

Differentiate e4x−7\displaystyle e^{4x-7}.

Show hint

The outer exponential stays unchanged.

Show worked solution
  1. Outer derivative is e4x−7\displaystyle e^{4x-7}.
  2. Inner derivative is 4.

Answer: 4e4x−7\displaystyle 4e^{4x-7}

4

Differentiate ln⁡(5x2+1)\displaystyle \ln(5x^2+1).

Show hint

Derivative of ln u is u'/u.

Show worked solution
  1. Inner derivative is 10x\displaystyle 10x.
  2. Divide it by the unchanged inner expression.

Answer: 10x/(5x2+1)\displaystyle 10x/(5x^2+1)

5

Differentiate 2x3−x\displaystyle \sqrt{2x^3-x}.

Show hint

Rewrite as a one-half power.

Show worked solution
  1. Outer derivative: 12(2x3−x)−1/2\displaystyle \frac12(2x^3-x)^{-1/2}.
  2. Inner derivative: 6x2−1\displaystyle 6x^2-1.

Answer: (6x2−1)/(22x3−x)\displaystyle (6x^2-1)/(2\sqrt{2x^3-x})

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

For a composition f(g(x))\displaystyle f(g(x)), differentiate the outer function while leaving the inner function unchanged, then multiply by the inner derivative: (f(g(x)))′=f′(g(x))g′(x)\displaystyle (f(g(x)))'=f'(g(x))g'(x).

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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