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A fundamental theorem concerning partial derivatives is that for a binary function f(x, y), providing that the derivatives exist,
The purpose of this article is to show that the analogous theorem concerning difference ratios also holds true. Let f(x, y) be a binary function. A difference ratio at (x, y) could be [f(x+h, y)−f(x−h, y)]/(2h). One cross difference ratio centered on (x, y) is
The other cross difference ratio is
Both of these reduce to
and thus the two cross difference ratios are equal.
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