= Solution
The <mixed partial derivative> $f_{xy}$ measures how the slope $f_y$ changes as $x$ varies while $y$ is fixed:
$$
f_{xy}(x,y)=\lim_{h\to0}\frac{f_y(x+h,y)-f_y(x,y)}h,
$$
when this limit exists. For a small rectangle, the cross-difference $f(x+h,y+k)-f(x+h,y)-f(x,y+k)+f(x,y)$ is approximately $hkf_{xy}(x,y)$ if the second derivatives are continuous. It measures a change of transverse slope, rather than either first slope alone. Mere continuous differentiability does not ensure that this second derivative exists; equality with $f_{yx}$ follows under the usual hypotheses of <Clairaut's theorem>.
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