Quotient rule: d/dx (u/v) = (v u′ − u v′) / v²
The quotient rule is d/dx (u/v) = (v du/dx − u dv/dx) / v².
What each letter means
- \(u\) the top (numerator), a function of x
- \(v\) the bottom (denominator), a function of x
When to use it
To differentiate one function divided by another, such as (2x + 1)/(x − 3) or sin x / x.
Worked example
Differentiate \(y=\dfrac{\sin x}{x}\).
- \(u=\sin x,\ v=x\); \(u'=\cos x,\ v'=1\)
Answer: \(\dfrac{dy}{dx}=\dfrac{x\cos x-\sin x}{x^2}\)
Common mistake
Swapping the order on top. It is v u′ − u v′; the other order flips the sign of the answer.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | In the Edexcel formulae booklet |
| Edexcel International A Level | In Pearson’s IAL formulae booklet |
| 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 quotient rule formula?
The quotient rule is d/dx (u/v) = (v du/dx − u dv/dx) / v². u: the top (numerator), a function of x; v: the bottom (denominator), a function of x.
Is the quotient rule given in the exam?
Edexcel A Level (9MA0): in the Edexcel formulae booklet. Edexcel International A Level: in Pearson’s IAL formulae booklet. 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.
Is there a way to remember the order?
Start with the bottom: 'v du minus u dv, all over v squared'.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.