Confusing derivative in $C([0,1])$

Question

I'm trying to show the existence and find the derivative of $$F(f)(s)=\int_0^scos(f(t)^2)dt$$ I'm struggling because this is not looking as nice with the method we used in class to find the derivative of $$G(f)(s)=\int_0^sf(t)^2dt$$ For G, we simply computed $$G(f+h)-G(f)=\int_0^s2f(t)h(t)dt+\int_0^sh(t)^2dt=\int_0^s2f(t)h(t)dt+o(\|h\|_{C([0,1])})$$ and saw $DG$ evaluated at h is the linear transformation $$\int_0^s2f(t)h(t)dt$$ Now using this technique for $F$ I get a $h$'s on the inside of cosines and sines, and this troubles me because I sort of think trigonometric functions are not linear transformations. I notice the for the derivative of G it kind of just amounted to taking the derivative under the integral, so I'm wondering if that is the case in general. This kind of problem is new to me, so any help would be fantastic. The follow-up question is to show $$FX=\{{F(f)|f\in C([0,1])}\}$$ is compact. It looks like I'm supposed to apply the Arzela-Ascoli Theorem or showing $FX$ and $FX'$ are bounded, but showing $FX$ is equicontinuous seems like it may be difficult here possible. Will it help to use that $f(x)$ is bounded because it is continuous on a compact interval?