How do I find this partial derivative I have the following function u(x,y) defined as:
$$u(x,y) = \frac {xy(x^2-y^2)}{(x^2+y^2)}$$ when x and y are both non zero,
and $u(0,0)=0$
I want to compute its partial derivative $u_{xy}$ at (0,0).
How do I do this?

$$\frac{\partial u}{\partial x}(0,0) = \lim_{t \to 0} \frac{u(0+t, 0) - u(0,0)}{t}
= \lim_{t \to 0} \frac{0 - 0}{t}=0.
$$
And we have 
$$\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)(0,0) = \lim_{t \to 0} \frac{\frac{\partial u}{\partial x}(0,0+t) - \frac{\partial u}{\partial x}(0,0)}{t}=\frac{-t}{t}=-1.$$
