TeX source:
\begin{array}{rcl}f'_{xy}(x,y) &=& -1\left(x+y^2\right)^{-1-1}\cdot 2y\\& & -2\cdot e^{2xy}-2ye^{2xy}\cdot 2x\\\\&=& -\dfrac{2y}{\left(x+y^2\right)^2}+\left(-2-4xy\right)e^{2xy}\\\end{array}