generalizations of the Leibniz ruleGeneralizedLeibnizRule2004-07-28 02:31:552007-04-04 22:59:05Theorem% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}For the derivative, the product rule
$$
(fg)' = f'g + fg'
$$
is known as the \emph{Leibniz rule}. Below are various ways it
can be generalized.
\subsubsection*{Higher derivatives}
Let $f,g$ be real (or complex)
functions defined on an open interval of $\sR$. If
$f$ and $g$ are $k$ times differentiable, then
$$
(fg)^{(k)} = \sum_{r=0}^k {k \choose r} f^{(k-r)} g^{(r)}.
$$
\subsubsection*{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
$$
\frac{d^n}{dt^n}\prod_{i=1}^rf_i(t) = \sum_{n_1+\cdots+n_r=n} {n \choose n_1,n_2,\ldots,n_r} \prod_{i=1}^r \frac{d^{n_i}}{dt^{n_i}}f_i(t).
$$
where ${n \choose n_1,n_2,\ldots,n_r}$ is the multinomial coefficient.
\subsubsection*{Leibniz rule for multi-indices}
If $f,g:\sR^n \to \sR$ are smooth functions defined on an
open set of $\sR^n$, and $j$ is a multi-index, then
$$ \partial^j(fg) = \sum_{i\le j} {j \choose i} \partial^i(f)\, \partial^{j-i}(g),$$
where $i$ is a multi-index.
\begin{thebibliography}{3}
\bibitem{Leibniz}
Leibniz, Gottfried W. {\it 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 \PMlinkexternal{digital library of the
Berlin-Brandenburg Academy}{
http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=01-misc/1&seite:int=184}.
\end{thebibliography}