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This is a basic question which holds true for the examples I checked. I wonder if there is a theorem/lemma which says the following:

Suppose $f(s,t)$ is a smooth function in two variables $s$, $t$ over the product interval $I\times I$.

Is the following true? $$\frac{d}{ds}\int_I f(s,t)dt = \int_I \frac{d}{ds} f(s,t)dt$$

Thanks.

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thanks for the answer--john – john Jun 28 '13 at 9:22

Yes, there is a lot of theorems about that, ranging from the Riemann integral to the Lebesgue one. See, as a starting point, this page

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