quotient rule

When one quantity is divided by another and both are changing — a population divided by an area to get density, or distance divided by time to get speed when both vary — you need a way to find how fast the ratio itself changes. The quotient rule does exactly that: it differentiates a fraction f/g where the top and bottom are each functions.

The rule is (f/g)' = (f'g - fg') / g^2. In words: bottom times derivative of the top, minus top times derivative of the bottom, all divided by the bottom squared. The order in the numerator matters because of the minus sign — swapping the two terms flips the sign and gives the wrong answer. A common memory aid is 'low d-high minus high d-low, over low squared'. It comes straight out of the product rule combined with the chain rule applied to f times g^(-1).

Two reminders. The rule only makes sense where g is not zero, since you're dividing by g^2. And you can often dodge the quotient rule entirely: a fraction like (3x)/x^2 is easier rewritten as 3x^(-1) and handled with the power rule, and any 1/g can be written as g^(-1). The quotient rule earns its keep when the top and bottom are both genuinely tangled functions.