Question A$2(3$ marks$)$

Consider the function

$f(x,y)=x^3+2xy+y^{2}sinx.$

Find the partial derivatives $f_x$ and $f_y$ everywhere.

Using your preferred differentiability test $($e$.$g$.,$ checking the continuity of partial derivatives in a neighborhood$),$ determine whether $f$ is differentiable at $(0,0).$

State clearly any theorems or results used in your conclusion.