$\frac{\partial z}{\partial x} = \sin \left(x y\right) + \left(x - y\right) \cos \left(x y\right) \cdot y = \sin \left(x y\right) + y \left(x - y\right) \cos \left(x y\right)$，
$\frac{\partial z}{\partial y} = - \sin \left(x y\right) + \left(x - y\right) \cos \left(x y\right) \cdot x = - \sin \left(x y\right) + x \left(x - y\right) \cos \left(x y\right)$，
则$x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y} =$$x \sin \left(x y\right) + x y \left(x - y\right) \cos \left(x y\right)$$- \left[- y \sin \left(x y\right) + x y \left(x - y\right) \cos \left(x y\right)\right] = \left(x + y\right) \sin \left(x y\right)$.