Theorem 3.2.1.
Let \(y = y(t)\) be a piecewise continuous, exponentially bounded function and assume that \(y'\) is also exponentially bounded. Then for large values of \(s\)
\begin{equation*}
{\mathcal L}(y')(s) = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0),
\end{equation*}
where \(Y(s)\) is the Laplace transform of \(y\text{.}\)