we have
\begin{equation*}
\mathscr{L}[f'(t)]=\int\limits_{0}^{\infty}e^{-st}f'(t)\,dt
\end{equation*}
\begin{equation*}
= \left.e^{-st}f(t)\right\vert_{0}^{\infty}-\int\limits_{0}^{\infty}-se^{-st}f(t)\,dt
\end{equation*}
\begin{equation*}
=0-f(0)+s\int\limits_{0}^{\infty}e^{-st}f(t)\,dt = sL[f(t)]-f(0)
\end{equation*}