# Quotient rule

The derivative of u over v is (u'v - uv') divided by v^2. The order of the top terms matters.

Maths (university) · Part 03: Differentiation · Card 6 of 10 · Last updated 2026-10-19

## Analogy

Low d-high minus high d-low, over low squared: a rhyme to keep the order straight.

## Example

**d/dx of x/(x + 1)**

u = x, u' = 1. v = x + 1, v' = 1.
Top: 1(x + 1) - x(1) = 1.
Result: 1 / (x + 1)^2.

## In practice

Use it for fractions with x on top and bottom, unless rewriting as a product is easier.

## Practice

### Flashcards

- **State the quotient rule.** (u/v)' = (u'v - uv') / v^2.

## Questions people ask

### Do I have to use the quotient rule for every fraction?

No. If the bottom is just a number, use the power rule. If it is easier, rewrite u over v as u times v to the power -1 and use the product and chain rules. Both ways give the same answer, so choose the less messy one.

### Why is there a minus sign in the quotient rule?

Increasing the denominator makes the fraction smaller, so its change counts against the result. That is the minus in u prime v minus u v prime. Swapping the order of the two terms would flip the sign and give a wrong answer.

## Related concepts

- [Power rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-4)
- [Product rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-5)
- [Chain rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-7)
- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)

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