Chain Rule Practice Problems With Step-by-Step Answers

Short answer: the chain rule differentiates a function inside another function: if y = f(g(x)), then dy/dx = f′(g(x)) · g′(x). In words: differentiate the outside, keep the inside unchanged, then multiply by the derivative of the inside.

How to spot a chain-rule problem

Ask: "Is there something other than plain x inside a bracket, a root, an exponent or a trig function?" If yes, you need the chain rule. sin x does not need it; sin(3x²) does. e^x does not; e^5x−1 does.

A reliable method is to name the layers before you differentiate:

1. Write the inside as u (for example u = 3x²).
2. Differentiate the outside as if u were x (d/du sin u = cos u).
3. Multiply by du/dx (6x).
4. Substitute u back: cos(3x²) · 6x = 6x cos(3x²).

If you are unsure whether the product or quotient rule is needed instead, read Chain rule, product rule or quotient rule? first.

Worked problems

Problem 1: y = (4x − 7)⁵

Outside: u⁵ → 5u⁴. Inside: u = 4x − 7 → u′ = 4.
dy/dx = 5(4x − 7)⁴ · 4 = 20(4x − 7)⁴.

Problem 2: y = √(x² + 1)

Rewrite: (x² + 1)^1/2. Outside → ½u^−1/2. Inside derivative: 2x.
dy/dx = ½(x² + 1)^−1/2 · 2x = x / √(x² + 1).

Problem 3: y = e^{sin x}

Outside: e^u → e^u. Inside: sin x → cos x.
dy/dx = cos x · e^{sin x}.

Problem 4: y = ln(3x² + 2x)

Outside: ln u → 1/u. Inside derivative: 6x + 2.
dy/dx = (6x + 2) / (3x² + 2x).

Problem 5 (three layers): y = sin³(2x)

Layers: cube of sine of 2x, i.e. [sin(2x)]³. Work outside in:

• Cube: 3[sin(2x)]²
• Sine: cos(2x)
• Inside: 2

dy/dx = 3 sin²(2x) · cos(2x) · 2 = 6 sin²(2x) cos(2x).

Problem 6 (chain + product): y = x² · cos(5x)

Product rule first: (x²)′cos(5x) + x²(cos 5x)′. The second derivative needs the chain rule: (cos 5x)′ = −5 sin(5x).
dy/dx = 2x cos(5x) − 5x² sin(5x).

Mistakes that cost marks

• Forgetting the inner derivative. d/dx sin(4x) is 4cos(4x), not cos(4x). This is the single most common error.
• Changing the inside. d/dx (x³ + 1)² is 2(x³ + 1) · 3x², not 2(3x²).
• Stopping one layer early on three-layer functions like sin³(2x).
• Misreading notation: sin²x means (sin x)², while sin(x²) has x² inside.

Practice section

Try these on paper before checking the answers below.

1. y = (2x + 3)⁶
2. y = cos(x³)
3. y = e^−4x
4. y = ln(cos x)
5. y = √(1 − x²)

Then test yourself with the 10-question quiz below. If you keep missing the same type of question, the free calculus diagnostic shows which topics are costing you marks. More background: derivatives topic page and calculus readiness check.