Laplace time differentiation property

[khdani]

Hello,
I'm trying to prove the time differentiation property of Laplace transform.
dx(t)/dt = sX(s)


http://img10.imageshack.us/img10/290/tlaplace.jpg http://g.imageshack.us/img10/tlaplace.jpg/1/

how do i continue from here ?

 

[Redbelly98]

I'm not sure what the first step here is supposed to represent. Wouldn't the first thing to do be to write

L[dx/dt] = ...
​

And then use the definition of the Laplace transform?

 

[khdani]

yes, you right
the first term should be like you said
but what i have to do with the first term of the last part?
it should vanish but i don't know how? maybe by Final Value Theorem ?

 

[Redbelly98]

In the final integral, t is the variable of integration.

Take any term that is independent of t outside of the integral, and see what you get.

p.s. your integration limits are wrong.

 

[khdani]

the final integral is sX(s)
but the one before should vanish somehow..., i think it's because of Final Value Theorem
why do you think the integration limits are wrong?

 

[Redbelly98]

the final integral is sX(s)

Yes, good.

... but the one before should vanish somehow..., i think it's because of Final Value Theorem
why do you think the integration limits are wrong?

It looks like there are two definitions of the Laplace transform:
http://mathworld.wolfram.com/LaplaceTransform.html

The integrations limits are usually taken from 0 to +∞, but maybe your class is using the -∞ to +∞ definition instead. I didn't know about that definition before.

If the limits are 0 to +∞, you can just evaluate the x(t) e^-st term at the limits. I'm not sure what to do if you're using the -∞ definition though.