# Laplace Transform of Derivative/Examples/Example 1

## Examples of Use of Laplace Transform of Derivative

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

$\laptrans {-3 \sin 3 t} = \dfrac {-9} {s^2 + 9}$

## Proof

 $\ds \laptrans {-3 \sin 3 t}$ $=$ $\ds s \laptrans {\cos 3 t} - \cos 0$ Laplace Transform of Derivative, Derivative of $\cos a x$ $\ds$ $=$ $\ds s \dfrac s {s^2 + 9} - 1$ Laplace Transform of Cosine, Cosine of Zero is One $\ds$ $=$ $\ds \dfrac {-9} {s^2 + 9}$ simplification

$\blacksquare$