Product rule: d/dx (uv) = u dv/dx + v du/dx
The product rule is d/dx (uv) = u dv/dx + v du/dx.
What each letter means
- \(u,\ v\) the two functions of x being multiplied
When to use it
To differentiate two functions multiplied together, such as x²eˣ or x sin x.
Worked example
Differentiate \(y=x^3\sin x\).
- \(u=x^3,\ v=\sin x\); \(\dfrac{du}{dx}=3x^2\), \(\dfrac{dv}{dx}=\cos x\)
Answer: \(\dfrac{dy}{dx}=x^3\cos x+3x^2\sin x\)
Common mistake
Multiplying the two derivatives. The derivative of uv is not u′v′.
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 product rule formula?
The product rule is d/dx (uv) = u dv/dx + v du/dx. u, v: the two functions of x being multiplied.
Is the product 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.
Can I expand the brackets instead?
If both parts are polynomials, yes: multiply out first. With eˣ, ln x or trig functions you need the product rule.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.