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Can one do this always, how to prove it? $$\frac{d}{dt} \int f(x,t) ~dx = \int \frac{df(x,t)}{dt} ~dx.$$

Why does it work on the wave-function in QM? I havent seen a requirement that it be continuous?

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Not always. In particular $$\int\limits_0^\infty {\frac{{\sin ax}}{x}dx} = \frac{\pi }{2}$$ for any $a$ so $$\frac{d}{{da}}\int\limits_0^\infty {\frac{{\sin ax}}{x}dx} = 0$$ but $$\int\limits_0^\infty {\frac{d}{{da}}\frac{{\sin ax}}{x}dx} = \int\limits_0^\infty {\cos axdx} = {\text{undefined}}$$ – Peter Tamaroff Jul 14 '12 at 21:30

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