Let $f(x) = f(c) + \int_c^x h(t) \ dt$ where $f,h:[a,b] \to \mathbb{R}$ and $h$ is riemann integrable, and $c \in [a,b]$. The book I'm looking at does this:
$$f'(x) = \frac{d}{dx}\left(f(c) + \int_c^x h(t) \ dt \right)$$ $$f'(x) = \frac{d}{dx}\left(f(c) + \int_a^x h(t) \ dt + \int_c^a h(t) \ dt \right)$$ $$f'(x) = h(x)$$
I am assuming they used the fundamental theorem of calculus, but I do not know how, or why the split of the integral was necessary