I am struggling to find whether there exists something like the Leibniz rule (interchanging integral and derivative) for Lebesgue-Stieltjes integrals. Does such a result exist? If so, how to use it?
In particular, how do we calculate
$\frac{d}{dx}\Big( \int_{a(x)}^{b(x)} f(x,t)dF(t) \Big)$?