Chapter 13.4, Problem 44E

Differentiability In Exercises 39 and 40, use the function to show that $f_x\left(0,0\right)$ and $f_y\left(0,0\right)$ both exist but that f is not differentiable at (0.0).

$f\left(x,y\right)=\{\begin{array}{cc}\frac{5x^{2}y}{x^3+y^3}& \left(x,y\right)\ne \left(0,0\right)\\ 0,& \left(x,y\right)=\left(0,0\right)\end{array}$