Fourier transform of a rect*half triangle I have to calculate the analytic expression of  Fourier transform  $$ x(t) = t{\rm rect} ( t- \frac{1}{2} ).$$ First I made the graph of these two signals and I obtained the graph I posted.

 

Now I wrote the analytic form of this graph that is $$ x(t) = t $$ for t from 0 to 1 and $$ x(t)=0 $$ for the rest. Now I should calculate the Fourier transform of $$ x(t) = t $$ but this should be $X(f)$ (?). My theory it’s completely wrong because my book obtained $$ X(f) = { \frac{1}{2} {\rm sinc} (f) + \frac{1}{i2\pi f}\left[({\rm sinc} (f )- \cos (\pi f ) \right]} e^{-i \pi f }. $$
There is a Fourier transform property called as "differenciation in frequency domain" which is as follows:

If the Fourier transform of $x(t)$ is $X(\omega)$, then the Fourier transform of $tx(t)$ is as below: $$\mathcal{F}(tx(t))  = j\frac{d}{d\omega} (X(\omega))$$

Where $\omega = 2\pi f$, 

I think you would be able to easily derive the answer you mentioned in your question now. 

Hope that helps