2.9 The Quotient Rule

The rule

For a quotient: d/dx[f/g] = (f'·g − f·g') / g². The numerator is 'derivative of top times bottom minus top times derivative of bottom,' all over the bottom squared.

Order matters

Because of the subtraction, order matters — you can't swap the terms. Example: d/dx[(sin x)/x] = (x·cos x − sin x)/x².

Key terms

Quotient rule

d/dx[f/g] = (f'g − fg')/g².

Numerator

f'g − fg' (order matters).

Denominator

The bottom function squared, g².

Order matters

Subtraction is not commutative.

Example

d/dx[(sin x)/x] = (x·cos x − sin x)/x².

Try an AP-style question

d/dx [x/(x + 1)] = ?

1. 1
2. 1/(x + 1)²
3. x/(x + 1)²
4. −1/(x + 1)²