Relating Transform of a Function and Transform of the Derivative

Please see the attached file for the fully formatted problems.

Problem statement:
What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will prove below is key to all that follows.

To do the following problem, you will need the definite integral form of the integration by parts formula:

Let be a function such that (you'll need this assumption in the following). Use integration by parts to show the following about Laplace transforms of derivatives.
Hint: makes an excellent .

OK, so I've said so far that
I let my u = and let dv = , du = -s and v = but then I get confused about how to implement the definite integral form of integration by parts, and won't I have to use this integration by parts several times?

After getting the integration by parts finished, I know I will have to take the limit as b approaches infinity from 0 to b. Then I can maybe apply L'Hopital's rule to simplify and prove that ?

Please see the attached file for the complete solution.
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Problem statement:
What really makes Laplace transforms work for differential equations is the relationship between the transform of a function and the transform of the derivative of that function. Therefore, the formula you will ...

Solution Summary

The Laplace transform of a function and its derivative are investigated. The solution is detailed and well presented.

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