Derivative of Laplace Transform

Theorem

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any closed interval $\closedint 0 a$.

Let $\laptrans f = F$ denote the Laplace transform of $f$.

Then, everywhere that $\dfrac \d {\d s} \laptrans f$ exists:

$\dfrac \d {\d s} \laptrans {\map f t} = -\laptrans {t \, \map f t}$

Proof

 $\displaystyle \frac \d {\d s} \laptrans {\map f t}$ $=$ $\displaystyle \frac \d {\d s} \int_0^{\to +\infty} \map f t \, e^{-s t} \rd t$ Definition of Laplace Transform $\displaystyle$ $=$ $\displaystyle \int_0^{\to +\infty} \map {\frac {\partial} {\partial s} } {\map f t \, e^{-s t} } \rd t$ Definite Integral of Partial Derivative $\displaystyle$ $=$ $\displaystyle \int_0^{\to +\infty} \map f t \, \map {\frac {\partial} {\partial s} } {e^{-st} } \rd t$ $\displaystyle$ $=$ $\displaystyle -\int_0^{\to +\infty} t \, \map f t \, e^{-s t} \rd t$ Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle -\laptrans {t \, \map f t}$ Definition of Laplace Transform

$\blacksquare$

Examples

Example 1

$\displaystyle \int_0^\infty t e^{-2 t} \cos t \rd t = \dfrac 3 {25}$