Solution
= Solution
Put $r=(x^2+y^2)^{1/2}$. Since $x^2y^2\leq(x^2+y^2)^2/4$, the nonzero-point expression satisfies
$$
0\leq |f(x,y)|\leq\frac{r^2}{4},\qquad\frac{|f(x,y)-f(0,0)|}{r}\leq\frac r4\longrightarrow0.
$$
Thus the defining remainder condition for <differentiability> holds with the zero <linear map>:
$$
\boxed{Df_{(0,0)}=0.}
$$
The estimate controls every approach direction at once, which is stronger than merely finding the two zero partial derivatives.