## Calculus (3rd Edition)

$$g_{xy}(-3,2)=24e^{6}.$$
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: $$g_{xy}(-3,2)=24e^{6}.$$