$$f(t) = t\cdot\sigma(t)e^{-t}, \text{where}\,\sigma(t)\text{ is the unit step function}$$
I thought about using the convolution theorem because we know that
$F[\sigma(t)e^{-t}](\omega) = \frac{1}{1+j\omega}$
However, I'm not sure what the transformation of $g(t) = t$ is, and so I'm not sure how to proceed. Also I've been trying to google it but I couldn't find anything.
I'd like to transform $f(t)$ without needing to derive any transformation, i.e. by using common transformations (and not the last one, that's too easy).