# Determining the derivative of an integral

1. Nov 27, 2009

### Calcgeek123

1. The problem statement, all variables and given/known data
Given: the integral from 0 to infinity of t^(x)e^(-t)dt
Problem: Determine f'(x).

2. Relevant equations

3. 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 thats 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.

2. Nov 27, 2009

### 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.

3. Nov 28, 2009

### ideasrule

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.

4. Nov 28, 2009

### Dick

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

5. Nov 28, 2009

### ideasrule

Oh. Sorry, I misunderstood the question.