Product rule

f

and

g

are differentiable at

a

, then

(f g)^{'} (a) = f^{'} (a) g (a) + f (a) g^{'} (a) .

Leibniz rule

The higher-order product rule is

(f g)^{(n)} = \sum_{k = 0}^{n} (k n) f^{(k)} g^{(n - k)} .

At zero,

f (x) = x

and

g (x) = 1 + ∣ x ∣

give a differentiable product

f (x) g (x) = x + x ∣ x ∣

with derivative one, although

g

is not differentiable.

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The product rule is a fundamental principle in calculus used to differentiate functions that are products of two (or more) functions.