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I would like to compute the integral

$$ \frac{\partial}{\partial{x}} \int_{-1}^1 \int_{-1}^1 F(x,y,\xi, \eta) \; d\xi \; d\eta $$

or moving the derivative inside the integrals

$$ \int_{-1}^1 \int_{-1}^1 \frac{\partial}{\partial{x}} F(x,y,\xi, \eta) \; d\xi \; d\eta $$

where $F(x,y,\xi, \eta)$ is a complicated rational function, the derivative of which is certain to result in expression swell.

I have come across only one example (a blog post containing an example) of automatic differentiation (AD) of an integral and I would like to see more, specially involving multiple integrals.

Olumide
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  • What's the problem with moving the derivative inside the integral? Are the assumptions of Leibniz's rule not satisfied? – Federico Poloni Mar 18 '22 at 08:40
  • @FedericoPoloni The problem, as I see it, is the complexity of $F(x,y, \xi, \eta)$. This makes the analytical derivative lengthier than $F(x,y, \xi, \eta)$, aka expression swell, which is one of the problems AD was designed to solve, in spite of the integral. – Olumide Mar 18 '22 at 09:25
  • You mention rational function. Is there any chance to use Feynman's parameter trick to obtain the integral before doing the derivative? – Bort Mar 18 '22 at 09:54

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