Functions of two variables and partial derivatives practice question 13

$f_{xy}$ means differentiate in $x$ first, then in $y$. At each stage, which factors of the function contain the variable you are differentiating in?

Find $f_x$ first. Both $x^{2}$ and $e^{xy}$ contain $x$, so use the product rule, with $\dfrac{\partial}{\partial x}e^{xy} = y\,e^{xy}$.

Then differentiate $f_x = e^{xy}\left(2x + x^{2}y\right)$ in $y$, holding $x$ fixed: the product rule again. As a check, $f_y = x^{3}e^{xy}$, and $f_{yx}$ must agree.