Product rule for higher partial derivatives
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 with respect to the first substring and with respect to the second substring.
- Add up all these terms. Since the number of substrings is (with the order of the derivative), we expect a sum of 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.
|second-order mixed partial derivative with respect to of a product of functions that involve at least two variables|
|derivative of the form of a product of functions that involve at least two variables|