Thread: 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?

If you have a function , with corresponding Laplace transform , then the Laplace transform of is , where the upper- and lower-case letters are very carefully typed. Similarly, the second derivative of would have the Laplace transform . 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.