I'm asked to show the following four derivatives from the polar coordinate conversions. ($x = r\cos \theta$, $y=r\sin \theta$, and $r^2=x^2+y^2$)
$$ \frac{\partial r}{\partial x} = \cos \theta \ , \ \frac{\partial r}{\partial y} = \sin\theta \ , \ \frac{\partial \theta}{\partial x} = \frac{-\sin \theta}{r} \ , \frac{\partial \theta}{\partial y} = \frac{\cos \theta}{r}$$
I've shown the fist two relatively easy, but I'm not sure how to show $$\frac{\partial \theta}{\partial x} = \frac{-\sin \theta}{r} \ , \frac{\partial \theta}{\partial y} = \frac{\cos \theta}{r}$$
I know how to do this for $$\frac{\partial x}{\partial \theta} \ \ \text{and} \ \ \frac{\partial y}{\partial \theta}$$
Any help would be appreciated, thank you.