Derivative of Laplace Transform

Theorem
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any closed interval $\left[{0 \,.\,.\, A}\right]$.

Let $\mathcal L \left\{{f \left({t}\right)}\right\}$ be the Laplace transform of $f$.

Then, everywhere that $\dfrac {\mathrm d \mathcal L}{\mathrm d s}$ exists:


 * $\dfrac {\mathrm d} {\mathrm ds} \mathcal L \left\{{f \left({t}\right)}\right\} = -\mathcal L \left\{{t \, f \left({t}\right)}\right\}$

Also see

 * Laplace Transform of Derivative