If $z = x^2 y + e^{xy^2}$, then $\left(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial x \partial y}\right)$ evaluated at $(1,0)$ is:}
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When computing partial derivatives that must be evaluated at a point where one of the coordinates is $0$, scan terms for high powers of that zero-variable early to eliminate tedious product rule computations. Here, since $y=0$, any term retaining an un-differentiated factor of $y$ drops immediately to zero.