Leibniz formula

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for the derivatives of a product

A formula that gives an expression for the $n$-th derivative of the product of two functions in terms of their derivatives of orders $k=0,\dots,n$ (the derivative of order zero is understood to be the function itself). Namely, if the functions $u=u(x)$ and $v=v(x)$ have derivatives up to the order $s$ inclusive at some point, then at this point their product $uv$ has derivatives of the same orders, and for $n=0,\dots,s$,

$$(uv)^{(n)}=\sum_{k=0}^n\binom nku^{(k)}v^{(n-k)}.$$

This formula was communicated by G. Leibniz in a letter to J. Bernoulli in 1695.

How to Cite This Entry:
Leibniz formula. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article