Use the function to show that $f_x(0,0)$ and $f_y(0,0)$ both exist, but that $f$ is not differentiable at $(0,0).$

$f(x,y)=\{\begin{array}{ccc}\frac{9x^{2}y}{x^4+y^2},& (x,y)(0,& 0)\\ 0,& (x,y)=(0,& 0)\end{array}$

$f_x(0,0)=$

$f_y(0,0)=$

Along the line $y=x_1,\underset{(x,y)(0,0)}{lim}=$

Along the curve $y=x^2,\underset{(x,y)(0,0)}{lim}=$

Therefore, $f(x,y)$ continuous at $(0,0)$ and hence $$ differentiable at $(0,0).$