# Thread: Laplace of the first and second derivatives

1. ## Laplace of the first and second derivatives

It's been a long time since I last used Laplace so please forgive me if this is a dumb question.

Assuming I take the first and second derivatives of a function should the Laplace transforms of those derivatives be the same?

2. If you have a function $f(t)$, with corresponding Laplace transform $F(s)$, then the Laplace transform of $f'(t)$ is $s F(s)-f(0)$, where the upper- and lower-case letters are very carefully typed. Similarly, the second derivative of $f(t)$ would have the Laplace transform $s^{2}F(s)-sf(0)-f'(0)$. So you gain two things by Laplace transformation: differentiation becomes multiplication, and the initial conditions are automatically included in your answer! Of course, not all functions even have a Laplace transform, so this method can only get you so far.