I was given the following exercise: knowing the Fourier transform of $g(x)=\frac{1}{1+x^2}$, compute the Fourier transform of $$f(x)=\frac{x}{(1+x^2)^2}$$
The problem is that maybe I don't know the useful property to solve it. The only way I know to "combine" known Fourier transforms to obtain a new one is convolution but I can't see how $f$ could be a convolution product. I also tried to apply the definition, without success.
Hint. One may recall that the Fourier transform of the derivative is given by $$ \mathcal{F}(f')(\xi)=2\pi i\xi\cdot\mathcal{F}(f)(\xi). $$ Check that $f$ is linked to the derivative of $g$.