Leibniz rule

In mathematics, the Leibniz rule is a rule for applying the nth power of a differential operator to a product function (differentiating n times the product function).

Let f(x) and g(x) be n times differentiable functions of x. Then Leibniz's rule states the following

${\frac {d^{n}{\big (}f(x)g(x){\big )}}{dx^{n}}}=\sum _{k=0}^{n}{\binom {n}{k}}\left({\frac {d^{k}f(x)}{dx^{k}}}\right)\left({\frac {d^{n-k}g(x)}{dx^{n-k}}}\right),$

where

${\binom {n}{k}}\equiv {\frac {n!}{(n-k)!k!}}$

is a binomial coefficient.