Consider the function

$f(x,y)=\{\begin{array}{ccc}\frac{x^3{y}^3}{x^2{y}^2},& (x,y)(0,& 0)\\ 0,& (x,y)=(0,& 0)\end{array}$

$($a$)$ Find $f_x(0,0)$ and $f_y(0,0).$

$($b$)$ Use the definition to prove that $f$ is not differentiable at $(0,0).$