Exercise 3, § 4

Let $I$ be an interval in $\mathbf{R}$, $f$ a mapping of $X \times I$ into a Banach space $F$, such that: $1^\circ$ for every $\alpha \in I$, the mapping $t \mapsto f(t, \alpha)$ of $X$ into $F$ is integrable; $2^\circ$ for every $t \in X$, the mapping $\alpha \mapsto f(t, \alpha)$ has a derivative $f'_\alpha(t, \alpha)$ in $I$; $3^\circ$ there exists an integrable function $g \geqslant 0$ such that $|f'_\alpha(t, \alpha)| \leqslant g(t)$ for all $t \in X$ and all $\alpha \in I$. Under these conditions, show that the function $u(\alpha) = \int f(t, \alpha) d\mu(t)$ is differentiable in $I$ and that