Leibniz’s theorem extends the product rule to determine higher-order derivatives of the product of two or more functions. It can be proven by mathematical induction.

Consider the function , where and are times differentiable. Using the product rule, the first few derivatives are:

which suggests that the -th order derivative of can be expressed as the binomial expansion

where and are non-negative integers, and are the -th order derivatives of and , respectively, and are the binomial coefficients.

Eq319a is known as Leibniz’s theorem, which can be proven by induction. For ,

If eq319a holds for all when , then for ,

Since (see this article for proof),

and the theorem holds for all and .