Differentiation under the integral $\frac{d}{dt}\int_0^s cos(f(x,t))\,dx$

Question

I'm trying for a while to solve this problem, but unfortunately without any success.

Assuming that the two functions $A$ and $f$ are known at time $t$ for the different positions $x$ (between $0$ and $1$). $f$ is subjected to Neumann's boundary conditions

$$ A(s,t) = \frac{d}{dt} \int_{0}^{s} \cos(f(x,t)) dx$$

How is it possible to determine the function $f(x,t+dt)$ at the next time from this equation?

Thanks in advance for your help.