San José State University
Thayer Watkins
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The Equality of the Cross Difference Ratios for a Binary Function

A fundamental theorem concerning partial derivatives is that for a binary function f(x, y), providing that the derivatives exist,

∂/∂y(∂f/∂x) = ∂/∂x(∂f/∂)
which is usually expressed as
∂²/∂y∂x = ∂²/∂x∂y

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

{[f(x+h, y+k)−f(x−h, y+k)]/(2h)−[f(x+h, y-k)−f(x−h,y-k)]/(2h)}/(2k)

The other cross difference ratio is

{[f(x+h, y+k)−f(x+h, y-k)]/(2k)−[f(x−h, y+k)−f(x−h,y-k)]/(2k)}/(2h)

Both of these reduce to

{f(x+h, y+k)−f(x−h, y+k)−f(x+h, y-k)+f(x−h,y−k)]}/(4hk)}

and thus the two cross difference ratios are equal.

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