Product rule for higher partial derivatives

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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_xg_y + f_yg_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_xg_{yy} + 2f_yg_{xy} + f_{yy}g_x + fg_{xyy}