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 \(y=x^2e^{3x}\).
- \(u=x^2,\ v=e^{3x}\); \(\dfrac{du}{dx}=2x\), \(\dfrac{dv}{dx}=3e^{3x}\)
Answer: \(\dfrac{dy}{dx}=3x^2e^{3x}+2xe^{3x}=xe^{3x}(3x+2)\)
Common mistake
Multiplying the two derivatives. The derivative of uv is not u′v′.
On your course
| Course | In the exam |
|---|---|
| AA SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA SL rules of differentiation questions
- Practise AA HL advanced differentiation questions
- Revise Differentiation (AA SL)
- Revise Differentiation (AA HL)
- Print AA SL 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?
AA SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.