Let
- $(\Omega,\mathcal A)$ be a measurable space
- $E,F$ be $\mathbb R$-Banach spaces
- $f:\Omega\times E\to F$ such that $f(\;\cdot\;,x)$ is strongly $(\mathcal A,\mathcal B(E))$-measurable for all $x\in E$ and $f(\omega,\;\cdot\;)\in C^1(E;F)$ for all $\omega\in\Omega$
Are we able to show the the Fréchet derivative ${\rm D}f(\;\cdot\;,x)$ of $f$ at $x\in E$ is strongly $(\mathcal A,\mathcal B(\mathfrak L(E,F)))$-measurable?
The claim is obviously true for the classical derivative in the case $E=F=\mathbb R$, since then ${\rm D}f(\;\cdot\;,x)$ is the limit of the measurable differentiation quotients.