For my current research, I need to prove the following:
\int_0^1 \frac{dC(q(x) + k'(q'(x) - q(x)))}{dk'}\,dk' = \int_0^1 \int_L^U p(q(x) + k(q'(x) - q(x)))(q'(x)-q(x)) dx dk
where C(q(x)) = \int_0^1 \int_L^U p(kq(x)) q(x)\,dx\,dk
Here's what I've tried using the definition of functional...