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I was watching this video on an intuitive proof of the Leibniz rule: https://www.youtube.com/watch?v=22JXq09-cqM . I can't understand the step where dt is moved outside of the integral sign. How and why does this happen?

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ten1o
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  • I haven't watched the video. This may help: https://math.stackexchange.com/questions/1991575/why-cant-the-second-fundamental-theorem-of-calculus-be-proved-in-just-two-lines/1991585#1991585 – Ethan Bolker Feb 14 '21 at 18:58

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From what I understand from the video, it's only a heuristic technique of computing the derivative of the integral $I(t) := \int_{x_1}^{x_2} f(x, t) dx.$ Thus, it is meant to be taken with a grain (or rather a full tablespoon) of salt. It is not rigorous. In his video, $dt$ signifies a finite quantity, however small, thus a constant, it is not to be taken as the $1-$form $dx$ (or measure, what have you, choose your preferred way of looking at integrals), thus can be safely moved outside the integral.

Actually Fritz
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