Product rule for higher partial derivatives

Statement

The general statement is a little complicated, because of the notation used for tracking the combinatorics. The idea is as follows:

• We consider the string with respect to which we are differentiating.
• Consider all ways of dividing this string into two substrings, each of which maintains the same order as the original string. One of the substrings may be empty.
• Now consider a term obtained by differentiating $f$ with respect to the first substring and $g$ with respect to the second substring.
• Add up all these terms. Since the number of substrings is $2^{n}$ (with $n$ the order of the derivative), we expect a sum of $2^{n}$ terms. In case there are repetitions in the string being differentiated with respect to, some of the terms may be equal and could be combined.

Particular cases

Case	Statement
second-order mixed partial derivative with respect to $x,y$ of a product $fg$ of functions that involve at least two variables $x,y$	$(fg)_{xy}=f_{xy}g+f_{x}g_{y}+f_{y}g_{x}+fg_{xy}$
derivative of the form ${}_{xyy}$ of a product $fg$ of functions that involve at least two variables $x,y$	$(fg)_{xyy}=f_{xyy}g+2f_{xy}g_{y}++f_{x}g_{yy}+2f_{y}g_{xy}+f_{yy}g_{x}+fg_{xyy}$