Change of Variables & the Jacobian · Superchallenge problem

Let $f(x,y) = \ln\!\left(e^{x} + e^{y} + e^{-x-y}\right)$. Evaluate $\displaystyle\iint_{\mathbb{R}^2}\left(f_{xx}f_{yy} - f_{xy}^2\right)dA$.