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:

$\displaystyle \left({f \left({x}\right) g \left({x}\right)}\right)^{\left({n}\right)} = \sum_{k \mathop = 0}^n \binom n k f^{\left({k}\right)} \left({x}\right) g^{\left({n - k}\right)} \left({x}\right)$

where $\left({n}\right)$ 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 $\alpha = \left({\alpha_1, \ldots, \alpha_n}\right)$ be a multiindex indexed by $\left\{{1, \ldots, n}\right\}$ with $\left\vert{\alpha}\right\vert \le k$.

For $i \in \left\{1,\ldots,n\right\}$ 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:

$\displaystyle \partial^\alpha\left({f g}\right) = \sum_{\beta \mathop \le \alpha} \binom \alpha \beta \left({\partial^\beta f}\right)\left({\partial^{\alpha - \beta} g}\right)$

Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.