# 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

\(\ds \frac \d {\d s} \laptrans {\map f t}\) | \(=\) | \(\ds \frac \d {\d s} \int_0^{\to +\infty} \map f t \, e^{-s t} \rd t\) | Definition of Laplace Transform | |||||||||||

\(\ds \) | \(=\) | \(\ds \int_0^{\to +\infty} \map {\frac {\partial} {\partial s} } {\map f t \, e^{-s t} } \rd t\) | Definite Integral of Partial Derivative | |||||||||||

\(\ds \) | \(=\) | \(\ds \int_0^{\to +\infty} \map f t \, \map {\frac {\partial} {\partial s} } {e^{-st} } \rd t\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds -\int_0^{\to +\infty} t \, \map f t \, e^{-s t} \rd t\) | Derivative of Exponential Function | |||||||||||

\(\ds \) | \(=\) | \(\ds -\laptrans {t \, \map f t}\) | Definition of Laplace Transform |

$\blacksquare$

## Examples

### Example 1

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

## Also see

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.10$ - 2009: William E. Boyce and Richard C. DiPrima:
*Elementary Differential Equations and Boundary Value Problems*(9th ed.): $\S 6.1$