 Solution

Inside the half-plane of convergence, the differentiation theorem for the Laplace transform permits differentiation with respect to the transform variable. Since ${\partial }_p{e}^{-pt}=-t{e}^{-pt}$,Repeating the argument, or applying induction using ${\partial }_p^n{e}^{-pt}=(-t{)}^n{e}^{-pt}$, givesThe identities hold for $\mathrm{Re}p>0$, within the stated convergence half-plane. The usual transform differentiation theorem uses an interior margin in the real part of $p$ to control the polynomial factors in $t$; it makes no assertion on the boundary $\mathrm{Re}p=0$.