My question is the same as this one: [$x =r\cos \theta$ and $y = r\sin\theta$, determine $\frac{\partial r}{\partial x}$ and $\frac{\partial \theta}{\partial x}$](https://math.stackexchange.com/questions/3365820/x-r-cos-theta-and-y-r-sin-theta-determine-frac-partial-r-partial-x?rq=1)

except that I do not know how to prove that $$ \frac{\partial \theta}{\partial x}=-\frac{\sin \theta}{r} $$ using the fact that $\tan \theta=\frac{y}{x}$.
Thanks for any help.