Suppose $I,J$ are compact intervals, $f:J\times I\rar E$ continuous, $u,v:J\rar I$ $C^1$ and $\pa_sf$ continuous. Then the function
$$
F(s)=\int_{v(s)}^{u(s)}f(s,t)\,dt
$$
is differentiable and
$$
F^\prime(s)=f(s,u(s))u^\prime(s)-f(s,v(s))v^\prime(s)
+\int_{v(s)}^{u(s)}\pa_sf(s,t)\,dt
$$
Define $h:(u,v,s)\mapsto\int_v^u f(s,t)\,dt$, then
$$
\pa_uh(u,v,s)=f(s,u),\quad
\pa_vh=-f(s,v),\quad
\pa_sh(u,v,s)=\int_v^u \pa_tf(s,t)\,dt
$$
and thus $h$ is $C^1$. Since $F(s)=h(u(s),v(s),s)$, we get
$$
F^\prime(s)
=\pa_uh(u(s),v(s),s)u^\prime(s)+\pa_vh(u(s),v(s),s)v^\prime(s)+\pa_sh(u(s),v(s),s)
$$