The question is taken from a Vector-Analysis worksheet as an extension exercise, here is the first part:
Let $\underline{v}$ be a vector field $\mathbb{R}^{2} \backslash (0,0)$ of the form
\begin{equation} \underline{v}(x,y) = f(r)(x,y) \end{equation}
where $r = \sqrt{x^{2}+y^{2}}$ and $f: (0,\infty) \to \mathbb{R}$ is continuously differentiable.
Find a scalar potential for $\underline{v}$ in terms of an integral involving $f$, given that $\underline{v}$ is conservative.
So far I have that since $\underline{v}$ is conservative, there exists a function $g(x,y)$ such that $\frac{\partial{g}}{\partial{x}} = xf(r)$ and $\frac{\partial{g}}{\partial{y}} = yf(r)$. Now I am a guessing that $g$ will in someway contain $F(t)$ where $F(t) = \int_0^rf(t)\,\mathrm{d}t$ since by the Fundamental Theorem of Calculus $F'(t) = f(r)$. However my knowledge of multivariable differentiation under the integral sign is not so good, and I am unsure how to take the partial derivatives of $F(t)$ with respect to $x$ or $y$. I am also unsure of the significance of $f$ being continuously differentiable. Any tips on where to go from here?