Fourier transform of a rect*half triangle
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