= Solution
The function $g(t)=|t|^{3/2}$ is <differentiable> everywhere: for $t\ne0$, its <derivative> is $(3/2)\operatorname{sgn}(t)\sqrt{|t|}$, while at zero the difference quotient is $\operatorname{sgn}(t)\sqrt{|t|}\to0$.
The function $k(t)=|t|^{1/2}$ is <differentiable> for $t\ne0$, with <derivative> $\operatorname{sgn}(t)/(2\sqrt{|t|})$. At zero its difference quotient is $\operatorname{sgn}(t)/\sqrt{|t|}$, which has no finite limit. Applying the coordinatewise-sum argument gives
$$
\boxed{f\text{ is differentiable exactly at }(x,y)\text{ with }y\ne0.}
$$
At those points,
$$
Df(x,y)(h,k)=\frac32\operatorname{sgn}(x)\sqrt{|x|}\,h+\frac{\operatorname{sgn}(y)}{2\sqrt{|y|}}\,k,
$$
where the first coefficient is zero at $x=0$. At any point with $y=0$, the restriction to the vertical coordinate line already fails to be <differentiable>.
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