Chain Rule Examples- Practice Problems and Solutions

What the Chain Rule Actually Is

The chain rule is a differentiation technique for composite functions. That's a fancy way of saying "functions inside other functions."

If you have f(g(x)), you differentiate the outer function, multiply by the derivative of the inner function. That's it. No magic, no mystery.

Most students either get this instantly or struggle for weeks. The difference? Practice. Pure and simple.

The Chain Rule Formula

For a composite function h(x) = f(g(x)):

h'(x) = f'(g(x)) × g'(x)

Or in Leibniz notation: if y = f(u) and u = g(x), then:

dy/dx = (dy/du) × (du/dx)

This second form is often easier to apply. You take the derivative of the outside function with respect to the inside, then multiply by the derivative of the inside.

Basic Chain Rule Examples

Example 1: Power of a Function

Find the derivative of f(x) = (3x + 1)⁵

The outer function is u⁵ and the inner function is u = 3x + 1.

Step 1: Differentiate the outer function → 5u⁴

Step 2: Multiply by the derivative of the inner → 5u⁴ × 3

Step 3: Substitute back → 15(3x + 1)⁴

That's your answer. No fluff needed.

Example 2: Trig Function with Chain Rule

Find the derivative of f(x) = sin(4x²)

Outer function: sin(u)

Inner function: u = 4x²

Derivative of sin(u) is cos(u). Derivative of 4x² is 8x.

Answer: f'(x) = 8x · cos(4x²)

Example 3: Exponential with Chain Rule

Find the derivative of f(x) = e^(x² + 3x)

The derivative of e^u is e^u. Multiply by the derivative of the exponent.

Derivative of x² + 3x is 2x + 3.

Answer: f'(x) = (2x + 3) · e^(x² + 3x)

Chain Rule with Multiple Applications

Some functions require the chain rule twice. Don't panic — just apply it twice.

Example: Find the derivative of f(x) = sin³(2x)

This is [sin(2x)]³. The outer function is u³, the inner is sin(2x).

Step 1: 3 · [sin(2x)]² · cos(2x) · 2

Step 2: 6 sin²(2x) cos(2x)

Notice the second application of the chain rule came from differentiating sin(2x) to get cos(2x) · 2.

Common Mistakes to Avoid

• Forgetting to multiply by the inner derivative. This is the #1 error. Every time.
• Overcomplicating simple problems. If it's just 5x³, you don't need the chain rule.
• Not simplifying at the end. Teachers expect simplified answers. Multiply those terms out.
• Confusing the product rule with the chain rule. Product rule is for two separate functions multiplied together. Chain rule is for nested functions.

Chain Rule vs Product Rule — Know the Difference

Practice Problems

Try these before checking the solutions. No peeking.

1. Find f'(x) if f(x) = (2x - 7)⁴
2. Find f'(x) if f(x) = √(5x + 2)
3. Find f'(x) if f(x) = cos(3x²)
4. Find f'(x) if f(x) = e^(5x)
5. Find f'(x) if f(x) = ln(x² + 4x)

Solutions to Practice Problems

Problem 1: f(x) = (2x - 7)⁴

Outer: u⁴, Inner: 2x - 7

4 · (2x - 7)³ · 2 = 8(2x - 7)³

Problem 2: f(x) = √(5x + 2)

Rewrite as (5x + 2)^(1/2)

(1/2) · (5x + 2)^(-1/2) · 5 = 5/(2√(5x + 2))

Problem 3: f(x) = cos(3x²)

Derivative of cos(u) is -sin(u). Derivative of 3x² is 6x.

f'(x) = -6x · sin(3x²)

Problem 4: f(x) = e^(5x)

Derivative of e^u is e^u. Derivative of 5x is 5.

f'(x) = 5e^(5x)

Problem 5: f(x) = ln(x² + 4x)

Derivative of ln(u) is 1/u. Derivative of x² + 4x is 2x + 4.

f'(x) = (2x + 4)/(x² + 4x)

You could factor the numerator to simplify further: 2(x + 2)/(x(x + 4))

Quick Reference: How to Apply the Chain Rule

1. Identify the outer function. Ask: what operation is applied last? That's your outer function.
2. Identify the inner function. Everything inside the outer function is the inner function.
3. Differentiate the outer function while keeping the inner function unchanged.
4. Differentiate the inner function.
5. Multiply the results.
6. Simplify.

Work through 20 problems and this process becomes automatic. Fewer than 20 and you're still guessing.