I am wondering whether the following result is true:
Let $F: [0, m] \rightarrow [0, 1]$ be a non-decreasing right-continuous function and $q: [0, m] \rightarrow \mathbb{R}_+$ be measuable. Suppose that the integral $\int_0^m q dF$ exists. Then, whenever $F$ is differentiable at $c \in [0, m]$, we have \begin{equation} \frac{d}{dc} \int_0^c q(z) dF(z) = q(c) F'(c). \end{equation}
Is there a reference which contains result? Thank you.