# Try to swap between mean and partial derivatives on a product

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• fab13
In summary, mean derivatives and partial derivatives serve different purposes in measuring the rate of change of a function. While mean derivatives give an overall view of a function's behavior over a specific interval, partial derivatives provide more specific information about the instantaneous rate of change with respect to a particular variable. Swapping between mean and partial derivatives can be useful in differentiating product functions with respect to different variables, and there are specific rules and formulas for doing so. Understanding mean and partial derivatives on a product can also be applied in various real-world scenarios, such as analyzing data and optimizing functions in fields like physics, economics, and engineering.

#### fab13

TL;DR Summary
I would like to be able to prove that we can swap the mean and partial derivatives on the defintion of a Fisher element matrix : this defintion involves the mean of a product of derivatives on Likelihood. I have also tried to formulate it with the ##chi^2## and the matrix of covariance of observables (noted "Cov" below). All of this is done in the goal that observable big "O" that I introduce is independent and so I have just to sum the extra elements calculated from "O".

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The operations are swappable if they're linear operations. Can you show that the operations are linear?

That said, I don't think this is a valid equation, as you seem to require, if my quick skimming of your post is correct:
$$\langle {\partial \chi^2 \over \partial \lambda_i \lambda_j} \rangle = \langle {\partial \chi^2 \over \partial \lambda_i} \rangle\langle {\partial \chi^2 \over \partial \lambda_j} \rangle$$