= Solution
{c}
For the <Laplace transform of a derivative>, first apply <integration by parts> on a finite interval:
$$
\int_0^T e^{-pt}\dot x(t)\,dt
=e^{-pT}x(T)-x(0)+p\int_0^T e^{-pt}x(t)\,dt.
$$
Under the usual transform hypotheses, including sufficient regularity for <integration by parts>, existence of the derivative's <Laplace transform>, and $e^{-pT}x(T)\to0$, passing to the <limit> gives
$$
\boxed{\mathcal L\{\dot x\}(p)=p\bar x(p)-x(0).}
$$
These 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 $\operatorname{Re}p>0$.
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