Chain rule: dy/dx = dy/du × du/dx
The chain rule is dy/dx = dy/du × du/dx.
What each letter means
- \(y\) a function of u
- \(u\) the inside function, a function of x
When to use it
To differentiate a function of a function: a bracket to a power, e to the power of an expression, sin or ln of an expression.
Worked example
Differentiate \(y=\ln(4x^2+1)\).
- Let \(u=4x^2+1\), so \(y=\ln u\)
- \(\dfrac{dy}{du}=\dfrac1u\), \(\dfrac{du}{dx}=8x\)
Answer: \(\dfrac{dy}{dx}=\dfrac{8x}{4x^2+1}\)
Common mistake
Forgetting to multiply by the derivative of the inside: d/dx (3x² + 1)⁵ is not just 5(3x² + 1)⁴.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Differentiation
- Revise IAL P3 further differentiation
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the chain rule formula?
The chain rule is dy/dx = dy/du × du/dx. y: a function of u; u: the inside function, a function of x.
Is the chain rule given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
How do I spot when to use the chain rule?
Look for one function inside another: a bracket raised to a power, eˢᵒᵐᵉᵗʰⁱⁿᵍ, sin(something) or ln(something), where the something is not just x.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.