Answer
$z_x=\sin{(xy)}+xy\cos{(xy)}$, $z_y=x^2\cos{(xy)}$.
Work Step by Step
$z=x\sin{(xy)}$
In order to find $z_x$ we treat $y$ as a constant and differentiate with respect to $x$.
$z_x=\sin{(xy)}+xy\cos{(xy)}$
In order to find $z_y$ we treat $x$ as a constant and differentiate with respect to $y$.
$z_y=x^2\cos{(xy)}$