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If we have $\sum_{i=1}^\infty f_i(x)$ and assume this is a convergent sum and asumme all the $f_i$ are differentiable in every point. Is the derivative of the infinite sum equal to the sum of the derivatives (so is $(\sum_{i=1}^\infty f_i(x))' = \sum_{i=1}^\infty f'_i(x)$ ) ?

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This is called "differentiation term-by-term". You can come up with examples where it fails. But in many cases it is correct. The best known case is the power series. In that case, differentiation term-by-term holds inside the radius of convergence.

Another one is due to Fubini ... if all $f_i$ are increasing, then term-by-term differentiation holds almost everywhere.

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