Leibniz's Rule

Theorem

One Variable

Let $f$ and $g$ be real functions defined on the open interval $I$.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are $n$ times differentiable.

Then:

$\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x$

where $\paren n$ denotes the order of the derivative.

Real Valued Functions

Let $f, g : \R^n \to \R$ be real valued functions, $k$ times differentiable on some open set $U \subseteq \R^n$.

Let $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ be a multiindex indexed by $\set {1, \ldots, n}$ with $\size \alpha \le k$.

For $i \in \set {1, \ldots, n}$, let $\partial_i$ denote the partial derivative:

$\partial_i = \dfrac {\partial} {\partial {x_i} }$

Let $\partial^\alpha$ denote the partial differential operator:

$\partial^\alpha = \partial_1^{\alpha_1} \partial_2^{\alpha_2} \cdots \partial_n^{\alpha_n}$

Then as functions on $U$, we have:

$\ds \map {\partial^\alpha} {f g} = \sum_{\beta \mathop \le \alpha} \binom \alpha \beta \paren {\partial^\beta f} \paren {\partial^{\alpha - \beta} g}$

Also known as

Leibniz's Rule is also known as Leibniz's theorem or Leibniz theorem.

Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.