$$
f(x,y) = \left\{
\begin{array}{ll}
\dfrac{xy}{\sqrt{x^2+y^2}} & \quad \text{if $x^2+y^2 \ne $ 0} \\
0 & \quad \text{if $x^2+y^2$ = 0}
\end{array}
\right.
$$
does its partial derivative exists at (0,0)
If I use the definition of partial derivatives, then I get answer as 0.
But if I first partially differentiate and then substitute values then it does not exist.