# The Laplace Transform of a Derivative

• January 29th 2009, 04:07 PM
razorfever
The Laplace Transform of a Derivative
show that $\mathcal{L} \left \{ \frac d{dt} f(t) \right \} = sF(s) - f(0)$

how???
• January 29th 2009, 09:46 PM
Jhevon
Quote:

Originally Posted by razorfever
show that L [df(t) / dt ] = sF(s) - f(0)

how???

By definition: $F(s) = \mathcal{L} \{ f(t) \} = \int_0^\infty e^{-st} f(t)~dt$

Thus, $\mathcal {L} \{ f'(t) \} = \int_0^\infty e^{-st} f'(t)~dt$

Now proceed using integration by parts with $u = e^{-st}$ and $dv = f'(t)~dt$