# Relating Transform of a Function and Transform of the Derivative

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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 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 ?

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

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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 ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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