This property enables solving differential equations algebraically. Derivatives in time become polynomials in \( s \), simplifying analysis.
\[ \mathcal{L}\left\{ \frac{d^n f(t)}{dt^n} \right\} = s^n \mathcal{L}\{f(t)\} - \sum_{r=0}^{n-1} s^{n-r-1} \left.\frac{d^r f(t)}{dt^r}\right|_{t=0} \]