 Solution

For the Laplace transform of a derivative, first apply integration by parts on a finite interval:Under the usual transform hypotheses, including sufficient regularity for integration by parts, existence of the derivative's Laplace transform, and $e^{-pT}x(T)\to 0$, passing to the limit givesThese conditions hold, for example, for a continuously differentiable $x$ with $x$ and $\dot{x}$ of exponential order, in a sufficiently far right half-plane. Existence of $\bar{x}(p)$ alone does not establish all of these derivative-transform hypotheses; the finite-interval identity displays exactly which boundary term must disappear. The polynomial solution in the last part satisfies them for every $\mathrm{Re}p>0$.