Answer
$$ g_{xy}(-3,2)=24e^{6}.$$
Work Step by Step
Since $ g(x,y)=xe^{-xy}$, then by using the chain rule, we have
$$ g_y(x,y)=\frac{\partial g}{\partial y}= -x^2e^{-xy}.$$
Using the product rule, we get
$$ g_{xy}(x,y)=\frac{\partial^2 g}{\partial x\partial y}= -2xe^{-xy}+x^2ye^{-xy}=(-2+x)xe^{-xy}.$$
Plugging in the coordinates gives:
$$ g_{xy}(-3,2)=24e^{6}.$$
