# Product rule

The derivative of u times v is u'v + uv'. You cannot just multiply the two derivatives.

Maths (university) · Part 03: Differentiation · Card 5 of 10 · Last updated 2026-10-19

## Analogy

A rectangle with growing sides: area changes because the width grows and because the height grows.

## Example

**d/dx of x^2 e^x**

u = x^2, u' = 2x. v = e^x, v' = e^x.
u'v + uv' = 2x e^x + x^2 e^x.
Factor: x e^x (2 + x).

## In practice

Use it whenever two non-constant factors, each with x in them, are multiplied.

## Practice

### Flashcards

- **State the product rule.** (uv)' = u'v + uv'.

### Interview questions

**Differentiate f(x) = x^2 sin(3x) and state which rules you used.**

- Product rule with u = x^2 and v = sin(3x)
- Power rule: u' = 2x
- Chain rule: v' = 3 cos(3x)
- Result 2x sin(3x) + 3x^2 cos(3x)

## Questions people ask

### Why is the derivative of a product not the product of derivatives?

Both factors change at once, and each change affects the total. Take x times x: the product of derivatives is 1, but the true derivative of x squared is 2x. The product rule adds both contributions: u prime v plus u v prime.

### How do I remember the product rule?

Say it in words: derivative of the first times the second, plus the first times derivative of the second. Differentiate one factor at a time while leaving the other alone, then add the results. It works for any two differentiable functions.

## Related concepts

- [Chain rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-7)
- [Power rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-4)
- [Quotient rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-6)
- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)

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