Let $x=r\cos\theta, y=r\sin\theta$. Consider a function $f(x,y)$.
Express $f_{xy}$ in polar form.
How should I proceed here? Should I differentiate $f(x,y)$ twice and find $f_{xy}$, and then replace $x$ and $y$ with $r\cos\theta$ and $r\sin\theta$? Or can I replace $x$ and $y$ with $r\cos\theta$ and $r\sin\theta$ in $f(x,y)$ itself and then differentiate with respect to $r$ and $\theta$?
Is $f_{xy}(x,y) = f_{r\theta}(r\cos\theta,r\sin\theta)=f_{\theta r}(r\cos\theta,r\sin\theta)$?