$$f(x,y)=\arctan \frac {x+y}{1-xy}.$$
So it is my intention to find out what second partial derivative $f_{xx}$, $f_{xy}$ and $f_{yy}$ are. But using quotient rule turns out to be complex.
Is there any easy way to calculate second partial derivative?
I'll add a caveat to Algin's excellent hint. The equality $$\arctan \frac {x+y}{1-xy}=\arctan(x)+\arctan(y)\tag{1}$$ is true when $xy<1$. When $xy>1$, the two sides differ by $\pi$: for example, with $x=2=y$ we have $\arctan (-4/3) = 2\arctan 2-\pi$. When $xy=1$, the left side of (1) is undefined.
The computation of derivatives is not affected much: $$f_{xx}(x,y)=(\arctan x)'', \ f_{yy}(x,y)=(\arctan y)'', \ f_{xy}(x,y)=0$$ provided that $xy\ne 1$. At the points with $xy=1$ the function $f$ is not continuous and does not have partial derivatives.
