Differentiation under the integral sign
= Differentiation under the integral sign
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If $g(t,x)$ and its partial derivative $D_i g(t,x)$ are continuous for $t$ in a compact interval and $x$ in an open set, then
$$
D_i\int_a^b g(t,x)\,dt
=\int_a^b D_i g(t,x)\,dt.
$$
Apply the <fundamental theorem of calculus> to the difference quotient in the $i$th coordinate and use uniform continuity on a compact neighbourhood.