If $z = e^{xy}$, calculate the value of $\frac{\partial^2 z}{\partial x \partial y}$ evaluated at $x = 1, y = 1$. (Give answer as a multiple of $e$, i.e. $\text{Value} / e$)
Answer Key:
2
Step-by-Step Solution & Explanation
$\frac{\partial z}{\partial y} = x e^{xy}$
$\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial}{\partial x}(x e^{xy}) = e^{xy} + x(y e^{xy}) = (1 + xy) e^{xy}$
At $x = 1, y = 1$:
$\frac{\partial^2 z}{\partial x \partial y} = (1 + 1) e^{1} = 2 e$
Value divided by $e$ is $2.0$.