Derivative of an improper integral

Question

Let $f(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t^2}dt$, $x \in (0,\infty)$. Show that $f(x)$ is differentiable in $(0,\infty)$ and find its derivative.

I don't know what direction I have to think to solve this. I think Leibniz rule of integration can be used but I am not able to apply that even.

Thanks in advance!

Answer 1

With $C^1$ nature of your function (Leibniz) you use this theorem

Hypothesis:

1) Your function in terms of $x$ is $C^1$

2) The derivative in terms of $x$ is at least continuous by part in terms of $t$

3) Domination hypothesis:

You can use a majoration of your function independent of x

$$\forall (x,t) \in {[0,+ \infty)}^2 $$

Or just for all a strictly positive ,on those intervals

$$ \forall (x,t) \in [a,+ \infty) \times {[1,+\infty)} $$

Answer 2

For example you have there :

1) and 2) obvious lets a positive

$$ \forall(x,t) \in [a,+\infty[\times[1,+\infty[, \ |\dfrac{e^{-xt}}{t}|\leq|\dfrac{e^{-at}}{t}| $$

it is integrable over $$ [1,+\infty[ $$

So you won.