General Leibniz rule

For other uses, see Leibniz's rule (disambiguation).

In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if $f$ and $g$ are $n$-times differentiable functions, then the product $fg$ is also $n$-times differentiable and its $n$th derivative is given by

$(fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(n-k)}g^{(k)},$

where ${n \choose k}={n! \over k!(n-k)!}$ is the binomial coefficient and $f^{(j)}$ denotes the jth derivative of f (and in particular $f^{(0)}=f$).

The rule can be proven by using the product rule and mathematical induction.

Second derivative

If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions:

$(fg)''(x)=\sum \limits _{k=0}^{2}{{\binom {2}{k}}f^{(2-k)}(x)g^{(k)}(x)}=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x).$

More than two factors

The formula can be generalized to the product of m differentiable functions f₁,...,f_m.