D4 (source code)
= D4
{c}
{scope}
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 $|\partial_tu(t,x)|$ uniformly there. Then $I'(t)=\int_E\partial_tu(t,x)\,dx$. 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.
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