How to prove the formula for L{tf(t)}?

[AdrianZ]

Homework Statement

L{tf(t)} denotes the Laplace transform of tf(t). I know that it's equal to -dF(s)/ds (where F(s)=L{f(t)}) from a Laplace transform table but I don't know how to prove that.

 

[micromass]

What is the definition of L{tf(t)}??

 

[AdrianZ]

What is the definition of L{tf(t)}??

$$\int^\infty_{0} tf(t)e^{-st} dt$$

I've tried integration by parts, but it didn't help.

 

[micromass]

OK, so I think it's best to start with

$$F(s)=\int_0^\infty{f(t)e^{-ts}dt}$$

Now we wish to find $F^\prime (s)$. Do you know Leibniz integral rule??

 

[AdrianZ]

OK, so I think it's best to start with

$$F(s)=\int_0^\infty{f(t)e^{-ts}dt}$$

Now we wish to find $F^\prime (s)$. Do you know Leibniz integral rule??

No. state the rule please.

 

[micromass]

http://en.wikipedia.org/wiki/Leibniz_integral_rule

But you'll going toi need something more general to deal with infinite regions.