Fourier transform of a function involving its derivative

Question

I am trying to understand how to go about the following:

We are given that the Fourier transform of a function $f(r)$ is $F(k)$ where $k$ is a representative wavenumber corresponding to some spatial coordinate $r$. If I am correct, then we can write:

$$F(k) = \int_{-\infty}^{\infty} f(r) e^{-ikr} dr$$

Now, how can one evaluate the Fourier transform of $ g(r) = \frac{d}{dr}(r f(r))$ in terms of $F(k)$? Any suggestions would be appreciated.

Answer 1

Using the properties of Fourier transforms the identity $g(r) = \frac{\mathrm{d}}{\mathrm{d} r}rf(r) $ becomes $$G(k) = - k\frac{\mathrm{d}}{\mathrm{d} k} F(k), $$ where $G$ is the Fourier transform of $g$.