 D4

A sufficient differentiation theorem is the following local version. Suppose $u(t,\cdot )$ is integrable, $u(\cdot ,x)$ is continuously differentiable for almost every $x$, and for each compact parameter interval there is an integrable $g(x)$ bounding $|{\partial }_{t}u(t,x)|$ uniformly there. Then $I^{\prime }(t)={\int }_E{\partial }_{t}u(t,x)dx$. The mean value theorem bounds the difference quotients by $g$, and dominated convergence proves the formula. Integrability at one parameter value together with the derivative bound also suffices locally.