# Leibniz formula

From Encyclopedia of Mathematics

*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: http://encyclopediaofmath.org/index.php?title=Leibniz_formula&oldid=33308

This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article