"Differentiating under the integral sign" theorem is well-known for $ f: I \times X \rightarrow\mathbb{R} \ $, such that $I$ is an interval of $\mathbb{R}$ , $ \left( X, \mathcal A, \mu \right) $ is a measure space and $f$ satisfies certain conditions. Is there a reference to a similar theorem if $I$ is replaced by an open set of a normed space and $f(.,x)$ is Fréchet-differentiable.
for example:
Let $ \left( X, \mathcal A, \mu \right) $ be a measure space and $ \left (E, \left\| \cdot \right \| \right )$ is a real normed space. Suppose $U \subset E$ is open and $ f: U \times X \rightarrow\mathbb{R} \ $ such that
$1)$ $ x \mapsto f(t,x) $ is a Lebesgue integrable function for all $t \in U$.
$2)$ $t \mapsto f(t,x)$ is a differentiable function in $U$ for a.e $x \in X$.
$3)$ There is an integrable function $g:X \rightarrow\mathbb{R_{+}}$ such that, for a.e $x \in X$ $$\left \| D_{t}f(\cdot,x) \right \|_{E^{'}} \leq g(x), \ \ \forall t \in U. $$
($D_{t}f(\cdot,x)$ denotes the Fréchet-differential of $f(\cdot,x)$ at $t$)
Then $ t \mapsto \int_X f(t,x) d\mu_{x} \ $ is differentiable in $U$ and for all $t \in U$ and $ h \in E $, we have $$ \ \ \left (D_{t} \left ( \int_X f(\cdot,x) d\mu_{x} \right ) \right )(h)= \int_X (D_{t}f(\cdot,x))(h) d\mu_{x} $$
Is there some book or textbook which cover theorems like this in English or German language?