Partial Derivatives & the Gradient · Challenge problem

Let $f(x,y) = \dfrac{xy(x^2 - y^2)}{x^2 + y^2}$ for $(x,y) \ne (0,0)$ and $f(0,0) = 0$. Compute $\dfrac{\partial}{\partial x}\!\left(\dfrac{\partial f}{\partial y}\right)\!(0,0) - \dfrac{\partial}{\partial y}\!\left(\dfrac{\partial f}{\partial x}\right)\!(0,0)$.