The Laplace transform for the derivatives of an appropiate function holds the relation: $$ \mathcal{L}\left(\frac{d^k}{dt^k}f(t)\right)(s) = s^k\mathcal{L}(f(t))(s) - \sum_{m=0}^{k-1}s^{(k-1)-m}\frac{d^m}{dt^m}f(t)\,\Bigg|_{t=0} $$
There is a formule for $$ \mathcal{L}\left(\frac{d^k}{dt^k}(t f(t))\right)(s)? $$
When developing this expression, the calculations are going to get very complicated, so I ask if there is any known method to do it.
Simply put $f(t)=tf(t)$. The formula will be: $$ -s^k \partial_s F(s)+\sum_{m=0}^{k-1}s^{(k-1)-m} (mf^{(m-1)}(0)) $$ where, $F(s)$ is the laplace transform of $f(t)$