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.