The similar question has already been asked, but there was no a comprehensive answer, so I'll ask it again.
I have a function that looks like:
$$f(x) = \int_{0}^{l}{F(t,x) dt},$$
where $F(t,x)$ is a continuous and differentiable function on intervals and on these intervals it can be represented like:
$$F(t,x) = a_1(t) * b_1(t+x) + a_2(t) * b_2(t+x),$$
where $a_1, a_2, b_1$ and $b_2$ are functions that can be expressed via some elementary functions. The question is: do I can express $${df(x)} \over {dx}$$ on these intervals, via functions $f$ or $F$, or any of $a_1,a_2,b_1,b_2$ functions, but without any integrals?
In addition: if we can't express the derivative, or the next one is easier to do, can we search for $f(x)$ function extreme values or its max/min values?
P.S.1: if it helps: function $F(t,x)$ is periodic function of the parameter $t$ on any $x$ value, with $l$ period.
P.S.2: sorry for bad English or bad question formulation.