Derivative of Laplace Transform

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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

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


Also see


Sources