= Solution
Write $F(p)=\mathcal L[f](p)$. Assume $f$ and $f'$ have right limits at zero and sufficient regularity and exponential-order bounds for the following <integration by parts>; take $\operatorname{Re}p$ large enough to kill the boundary terms at infinity. First,
$$
\mathcal L[f'](p)=[e^{-pt}f(t)]_0^\infty+pF(p)=pF(p)-f(0+).
$$
Apply the same calculation to $f'$:
$$
\boxed{\mathcal L[f''](p)=p^2F(p)-pf(0+)-f'(0+).}
$$
This is the <Laplace transform of a derivative> formula with both initial-data terms retained.
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