Higher Order Partial Derivatives — Question 8

Question 8 – Solution

Strategy. Restrict first derivatives to an axis, then differentiate that restriction at zero.

Step 1: Axis first partials For $y\ne 0$, $$f_x(0,y)=\lim_{h\to 0}\frac{h^2y^2/(h^2+y^2)}h =\lim_{h\to 0}\frac{hy^2}{h^2+y^2}=0.$$ At $y=0$ it is also zero. By symmetry, $f_y(x,0)=0$ for every $x$.

Step 2: Mixed values $$f_{xy}(0,0)=\lim_{k\to 0}\frac{f_x(0,k)-f_x(0,0)}k=0,$$ $$f_{yx}(0,0)=\lim_{h\to 0}\frac{f_y(h,0)-f_y(0,0)}h=0.$$ Thus $\boxed{f_{xy}(0,0)=f_{yx}(0,0)=0}$.

Verification. Each definition uses only the relevant axis restriction, so off-axis values do not enter these two particular quotients.