Product and quotient rules

The product and quotient rules are the standard tools for differentiating a function that is the product or quotient of two simpler functions. The product rule says that if h(x) = f(x) · g(x), then h′(x) = f′(x)g(x) + f(x)g′(x) — differentiate each factor in turn, leaving the other alone, and add. Crucially, the derivative of a product is not the product of the derivatives.

The quotient rule handles h(x) = f(x) / g(x): h′(x) = (f′(x)g(x) − f(x)g′(x)) / g(x)². A common memory aid uses "low d-high minus high d-low, over low squared." Because the numerator involves subtraction, order matters — swapping the terms flips the sign of the answer.

A quick example of each. For h(x) = x² sin x, the product rule gives h′(x) = 2x sin x + x² cos x. For h(x) = sin x / x, the quotient rule gives h′(x) = (x cos x − sin x) / x². Any quotient can also be rewritten as a product with a negative exponent and differentiated with the product and chain rules, which is a useful check.

These rules are core content on the AP Calculus AB exam, where they combine with the chain rule in nearly every differentiation problem, and they appear on the FE Mechanical exam within the mathematics topics. Expect to apply them to polynomial, trigonometric, exponential, and logarithmic combinations, and to recognize when a product or quotient structure requires them.