find the derivative of the integral Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative.
$$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$
I don't know how to get rid of the absolute value in the integral
any ideas?

$\dfrac{dF(x)}{dx}$
$=\dfrac{d}{dx}\int_0^1e^{|x-y|}~dy$
$=\dfrac{d}{dx}\int_0^1e^{(x-y)\text{sgn}(x-y)}~dy$
$=\int_0^1\text{sgn}(x-y)e^{(x-y)\text{sgn}(x-y)}~dy$
$=[-e^{(x-y)\text{sgn}(x-y)}]_0^1$
$=[-e^{|x-y|}]_0^1$
$=e^{|x|}-e^{|x-1|}$
