Determining the derivative of an integral

[Calcgeek123]

Homework Statement

Given: the integral from 0 to infinity of t^(x)e^(-t)dt
Problem: Determine f'(x).

Homework Equations

The Attempt at a Solution

My teacher mentioned using the definition of a derivative:
f'(a)= limit as x approaches a of f(x)-f(a)/(x-a).

So far I have: f'(a)=the integral of the limit as x approaches a of t^(x)e^(-t)-t^(a)e^(-t)/(x-a) dt.

I'm not sure where to go from here, or if that's even correct. I think it should end up being f'(x)=t^(x)e^(-t) which makes sense to me. I'm just not sure how to actually get there.

Thank you to all.

 

[Dick]

Look at the Leibniz rule for differentiation of an integral. If you actually want to do the resulting integral, you'll need to use a gamma function, but maybe you can just give the answer in terms of another integral.

 

[ideasrule]

Homework Statement

Given: the integral from 0 to infinity of t^(x)e^(-t)dt
Problem: Determine f'(x).

I don't quite understand this. What's the x in t^(x)e^(-t)dt? Is it just a constant?

Anyhow, the integral from 0 to infinity of that function should be a constant, like 2 or 2.313. Deriving a constant gives you 0.

 

[Dick]

I don't quite understand this. What's the x in t^(x)e^(-t)dt? Is it just a constant?

Anyhow, the integral from 0 to infinity of that function should be a constant, like 2 or 2.313. Deriving a constant gives you 0.

Changing the value of x changes the value of the integral. Try x=0, x=1, etc. It's not a constant.

 

[ideasrule]

Oh. Sorry, I misunderstood the question.