Differentiation under the Integral Sign

Let $(X,\mathcal{F},\mu)$ be a measure space, let $p$ and $q$ be real numbers with $p<q$, and let $U$ be the open interval with endpoints $p$ and $q$. By An Open Interval is an Interval All of Whose Points Are Interior every point of $U$ is an interior point of $U$, so differentiability at a point of $U$ is meant for functions defined on $U$. Let $f:U\times X\to\mathbb{R}$ satisfy the following three conditions.

(i) For every $t\in U$ the function $x\mapsto f(t,x)$ is integrable with respect to $\mu$.

(ii) For every $x\in X$ the function $s\mapsto f(s,x)$ is differentiable at every point of $U$; its derivative at $t\in U$ is written $D_1f(t,x)$.

(iii) There is an integrable function $g:X\to\mathbb{R}$ such that $|D_1f(t,x)|\le g(x)$ for every $t\in U$ and every $x\in X$.

Then for every $t\in U$ the function $x\mapsto D_1f(t,x)$ is measurable and integrable, the function $F:U\to\mathbb{R}$ given by

$$F(t)=\int_X f(t,x)\,d\mu(x)$$

is differentiable at every point of $U$, and

$$F'(t)=\int_X D_1f(t,x)\,d\mu(x)$$

for every $t\in U$.