Answer
$f$ is not differentiable in $(x,y)=(0,0)$
Work Step by Step
In order to find the partial derivative, we will differentiate with respect to $x$, by keeping $y$ as a constant, and vice versa:
$f_x=\dfrac{\partial }{\partial x}(\dfrac{xy^2)}{x^2+y^4})=\dfrac{y^2(x^2+y^4)-xy^2(2x+y^4)}{(x^2+y^4)^2}=\dfrac{y^6-x^2y^2-xy^6}{(x^2+y^4)^2}$
and $f_y=\dfrac{\partial }{\partial y}(\dfrac{xy^2)}{x^2+y^4})=\dfrac{2xy(x^2+y^4)-xy^2(x^2+4y^3)}{(x^2+y^4)^2}=\dfrac{2x^3y-x^3y^2-2xy^5}{(x^2+y^4)^2}$
Now, $\lim\limits_{(y^2,y) \to (0,0)} f_x=\lim\limits_{(y^2,y) \to (0,0)}\dfrac{y^6-y^6-y^8}{(y^4+y^4)^2}=\dfrac{-1}{4}$
and $\lim\limits_{(-y^2,y) \to (0,0)} f_y=\lim\limits_{(-y^2,y) \to (0,0)}\dfrac{y^6-y^6-y^8}{(y^4+y^4)^2}=\dfrac{1}{4}$
Thus, we conclude that $f_x$ and $f_y$ are not continuous in $(0,0)$, so $f$ is not differentiable in $(x,y)=(0,0)$.
