Let $f(x,y)$ be a smooth function on $\mathbb{R}^2$. Suppose that its gradient vector field $$ \nabla f:=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right) $$ is invariant under rotations about the origin of $\mathbb{R}^2$. (Notice that we do not assume a priori that $f$ is invariant under rotations.)
Under this assumption, how to prove that $\nabla f$ is proportional to the radius vector $(x,y)$?