# generalizations of the Leibniz rule

For the derivative, the product rule

 $(fg)^{\prime}=f^{\prime}g+fg^{\prime}$

is known as the Leibniz rule. Below are various ways it can be generalized.

## Higher derivatives

Let $f,g$ be real (or complex) functions defined on an open interval of $\mathbb{R}$. If $f$ and $g$ are $k$ times differentiable, then

 $(fg)^{(k)}=\sum_{r=0}^{k}{k\choose r}f^{(k-r)}g^{(r)}.$

## Generalized Leibniz rule for more functions

Let $f_{1},\ldots,f_{r}$ be real (or complex) valued functions that are defined on an open interval of $\mathbb{R}$. If $f_{1},\ldots,f_{r}$ are $n$ times differentiable, then

 $\frac{d^{n}}{dt^{n}}\prod_{i=1}^{r}f_{i}(t)=\sum_{n_{1}+\cdots+n_{r}=n}{n% \choose n_{1},n_{2},\ldots,n_{r}}\prod_{i=1}^{r}\frac{d^{n_{i}}}{dt^{n_{i}}}f_% {i}(t).$

where ${n\choose n_{1},n_{2},\ldots,n_{r}}$ is the multinomial coefficient.

## Leibniz rule for multi-indices

If $f,g:\mathbb{R}^{n}\to\mathbb{R}$ are smooth functions defined on an open set of $\mathbb{R}^{n}$, and $j$ is a multi-index, then

 $\partial^{j}(fg)=\sum_{i\leq j}{j\choose i}\partial^{i}(f)\,\partial^{j-i}(g),$

where $i$ is a multi-index.

## References

• 1 Leibniz, Gottfried W. Symbolismus memorabilis calculi Algebraici et Infinitesimalis, in comparatione potentiarum et differentiarum; et de Lege Homogeneorum Transcendentali, Miscellanea Berolinensia ad incrementum scientiarum, ex scriptis Societati Regiae scientarum pp. 160-165 (1710). Available online at the http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=01-misc/1&seite:int=184digital library of the Berlin-Brandenburg Academy.
Title generalizations of the Leibniz rule GeneralizationsOfTheLeibnizRule 2013-03-22 14:30:18 2013-03-22 14:30:18 GeraW (6138) GeraW (6138) 13 GeraW (6138) Theorem msc 26A06 Leibniz rule MultinomialTheorem NthDerivativeOfADeterminant