#### Answer

$$z_x = \sin{(xy)} + x y \cos{(xy)}.$$
$$z_y = x^2 \cos{(xy)}.$$

#### Work Step by Step

To find $z_x$, we treat $y$ as a constant and differentiate with respect to $x$. Using the product rule, we have
$$z_x = 1 \cdot \sin{(xy)} + x \cdot y \cos{(xy)}.$$
Similarly, to find $z_y$, we treat $x$ as constant and differentiate with respect to $y$. Thus
$$z_y = x \cdot x \cos{(xy)} = x^2 \cos{(xy)}.$$