I have the following function and I want to show that it is differentiable. I am going to do this by showing that the partial derivatives are continuous and so I will show that they are continuous at (0,0). So, i am going to show that as the limit of (x,y) approaches (0,0) the derivative approaches 0.
$$f(x,y)=\left\{\begin{array}{l} \frac{x^2y^2}{\sqrt{x^2+y^2}},\:\text{if $(x,y) \not= (0,0)$;}\\ 0,\:\text{if $(x,y)=(0,0)$;} \end{array}\right.$$
$$\begin{cases} \dfrac {\partial f}{\partial x}(x,y)= \dfrac{x^3y^2+2xy^4}{({x^2+y^2})^\frac{3}{2}}\\ \dfrac {\partial f}{\partial x}(0,0)=0\end{cases}$$
I am having trouble however and was wondering if anyone could work me through this case.