Partial Derivative which when evaluated gives $0/0$ My function is $f(x,y)=xy\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ EDIT defined $f(0,0)=0$ (How does that impact the evaluation of partial derivatives?)

I need to evaluate the function at $f_{xy}(0,0)$ and similarly for $f_{yx}(0,0)$ My problem is that upon evaluating the function at (0,0), I get cases of $0/0$ How can I get around it? Should I try rearranging terms with algebra?
You'll need to use the definition of derivative to find $f'_x$ and $f'_y$ at $(0,0)$; for $(x,y)\neq (0,0)$ you just use differentiate the expression for $f$ as usual. Then you have formulas $f'_x(x,y)$ equal to something at $(0,0)$ and something else elsewhere, and similarly for $f'_y(x,y)$. Then you again use the definition of derivative on those functions to get $f''_{xy}$ and $f''_{yx}$ at $(0,0)$.
Let $y=ax$ and take the limit as $x \rightarrow 0$.

Edit: If the result of taking the above limit is not a constant but a function of $a$, the limit does not exist. You can, however, find the value of $f_{xy}(0,0)$ or $f_{yx}(0,0)$ from a particular direction by picking $a$ such that $y=ax$ defines the line along which you want to approch $(0,0)$.