= Solution
A sufficient theorem is <dominated differentiation under the integral sign>. Let $t_0$ be interior to a parameter interval, suppose $u(\cdot,t)$ is measurable for each $t$, and suppose $u(\cdot,t_0)$ is integrable. Assume $u(x,\cdot)$ is differentiable near $t_0$ for $x$ outside one fixed <null set>, and that throughout this neighborhood
$$
|\partial_tu(x,t)|\leq H(x)
$$
for one nonnegative <Lebesgue integrable function> $H$. Then $F(t)=\int_Eu(x,t)\,dx$ is defined near $t_0$, is differentiable at $t_0$, and
$$
\boxed{F'(t_0)=\int_E\partial_tu(x,t_0)\,dx.}
$$
Indeed the <mean value theorem> bounds both $|u(x,t)-u(x,t_0)|$ by $|t-t_0|H(x)$ and the difference quotient by $H(x)$. The first bound proves local integrability; the second lets the <dominated convergence theorem> pass the limit of difference quotients through the integral.
Back to article page
