Higher Order Partial Derivatives — Question 6

Question 6 – Solution

Strategy. Apply one-variable mean value arguments successively across the small rectangle.

Step 1: Difference structure Let $$G(y)=f(a+h,y)-f(a,y).$$ The numerator is $G(b+k)-G(b)$.

Step 2: Mean value steps For some $\eta$ between $b$ and $b+k$, $$G(b+k)-G(b)=kG'(\eta) =k\bigl[f_y(a+h,\eta)-f_y(a,\eta)\bigr].$$ For some $\xi$ between $a$ and $a+h$, the bracket equals $h f_{yx}(\xi,\eta)$. Hence $$D(h,k)=f_{yx}(\xi,\eta).$$

Step 3: Limit As $(h,k)\to(0,0)$, $(\xi,\eta)\to(a,b)$. Continuity and equality of mixed partials give $$\boxed{\lim D(h,k)=f_{yx}(a,b)=f_{xy}(a,b)}.$$