The chain rule differentiates a bracket raised to a power. Work from the outside in: differentiate the power, keep the bracket, then multiply by the derivative of the inside.
This lesson is part of SPM Additional Mathematics differentiation. It builds on differentiating powers, products and quotients.
What does the rule say?
If y = (f(x))ⁿ, then dy/dx = n (f(x))ⁿ⁻¹ × f′(x). The last factor is the derivative of the inside.
Another way to write it uses u for the inside: if y is a function of u and u is a function of x, then dy/dx = dy/du × du/dx.
Worked example 1: a bracket to a power
Differentiate y = (3x + 2)⁵.
Outside: the power 5 comes down and the new power is 4, giving 5(3x + 2)⁴. Inside: the derivative of 3x + 2 is 3.
Multiply: dy/dx = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴.
Worked example 2: a square root
Differentiate y = √(4x − 1).
Rewrite as y = (4x − 1)^(1/2). Outside: (1/2)(4x − 1)^(−1/2). Inside: the derivative of 4x − 1 is 4.
So dy/dx = (1/2)(4x − 1)^(−1/2) × 4 = 2(4x − 1)^(−1/2), which is 2 ÷ √(4x − 1).
Worked example 3: using u
Differentiate y = (x² + 1)³ by setting u = x² + 1.
Then y = u³, so dy/du = 3u², and du/dx = 2x. Multiply: dy/dx = 3u² × 2x = 6x(x² + 1)².
To find the gradient at x = 1, put x = 1: 6(1)(2)² = 24.
The mistake that costs marks
The common slip is to differentiate the outside and forget the inside.
| Question | Wrong | Right |
|---|---|---|
| d/dx of (2x + 1)⁴ | 4(2x + 1)³ | 4(2x + 1)³ × 2 = 8(2x + 1)³ |
A quick check exposes it. Take (2x + 1)² = 4x² + 4x + 1, which differentiates directly to 8x + 4 = 4(2x + 1).
The chain rule gives 2(2x + 1) × 2 = 4(2x + 1), which matches. Leaving out the inner factor 2 would give 2(2x + 1), exactly half of the correct answer.
Check yourself
Differentiate y = 1 ÷ (2x − 5)², then find the gradient at x = 3.
Answer
Rewrite as y = (2x − 5)^(−2).
dy/dx = −2(2x − 5)^(−3) × 2 = −4(2x − 5)^(−3) = −4 ÷ (2x − 5)³.
At x = 3: 2x − 5 = 1, so the gradient is −4 ÷ 1 = −4.
What to study next
Apply the chain rule to gradients in finding tangents and normals. Practise the rules with the polynomial differentiation tutor, then test yourself on the differentiation practice set.
If you want a teacher to watch where the inner derivative goes missing, see online one-to-one Additional Mathematics tuition.