The Lagrangian function of the problem (12) is expressed as $$ \mathcal L = \frac{W}{2v}\int_x^y\log_2\left(1 + \frac{p(s)\beta}{(s^2 + H^2)^{\alpha/2}}\right)\mathrm ds-\lambda\left(\frac1v\int_x^yp(s)\mathrm ds - E\right)\tag{35} $$ By setting $\frac{\partial\mathcal L}{\partial p(s)} = 0$, we get the optimal power allocation [...]
This is a derivative problem in a paper, but I have no idea about how to do a derivative respect to a function in the definite integral.
I'm no expert on Lagrangian mechanics, but I've come across it on a few occations. I leave a few thoughts here in something which may or may not be an answer to your question.
This is basically the main idea behind Lagrangian mechanics: Given a few initial conditions (like what the value of $p$ is at the end points of the integral), find the functions $p$ which minimise $\mathcal L$. Then if $\mathcal L$ is formulated correctly, such a $p$ will be a valid path the system can take.
Just as you're used to with finding maxima and minima in school, if $p$ is indeed a minimum of $\mathcal L$, a tiny change in $p$ induces a comparably much smaller change in the value of $\mathcal L$. In true physics fashion this is usually just written as $\frac{\partial L}{\partial p}$.
It is difficult to find such a $p$ directly, but thankfully, the Euler-Lagrange equations will give you a differential equation that any such $p$ must obey (assuming it is twice differentiable, and all the other implicit niceness assumptions physics likes to make). That's where the actual calculations to find the right $p$ begins.