How do I start solving this question

Question

I thought really hard about this question but I can't figure out how to start solving it. If it was given what $f$ was I could use $x=r\cos(\theta)$ and $y=r\sin(\theta)$ to prove it.

Thanks for any help.

Answer 1

Since $f$ is a function of $x,y$ and each of $x,y$ is a function of $r,\theta$, the chain rule says that

\begin{align*} \frac{\partial f}{\partial r} & = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial r}\\ & = f_x \cdot \frac{\partial}{\partial r}(r \cos(\theta)) + f_y \cdot \frac{\partial}{\partial r}(r \sin(\theta)) \\ & = f_x \cdot \cos(\theta) + f_y \cdot \sin(\theta) \end{align*}