Past exam of the mathematical methods II course of the University of Oxford / 2017 / 11 / a / Solution

The mixed partial derivative $f_{xy}$ measures how the slope $f_y$ changes as $x$ varies while $y$ is fixed: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 $hk{f}_{xy}(x,y)$ 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.