Try to swap between mean and partial derivatives on a product

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.
  • #1
fab13
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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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  • #2
The operations are swappable if they're linear operations. Can you show that the operations are linear?
 
  • #3
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:
$$\langle {\partial \chi^2 \over \partial \lambda_i \lambda_j} \rangle = \langle {\partial \chi^2 \over \partial \lambda_i} \rangle\langle {\partial \chi^2 \over \partial \lambda_j} \rangle$$
 

1. What is the difference between mean and partial derivatives?

Mean derivatives refer to the average rate of change of a function over a given interval, while partial derivatives refer to the rate of change of a multivariable function with respect to one of its variables, holding all other variables constant.

2. When would you need to swap between mean and partial derivatives on a product?

You would need to swap between mean and partial derivatives on a product when dealing with a multivariable function that involves a product of multiple variables. This allows you to analyze the rate of change of each variable separately.

3. How do you calculate the mean derivative of a product?

To calculate the mean derivative of a product, you would first need to take the derivative of the product using the product rule. Then, you would divide the resulting derivative by the length of the interval over which the mean derivative is being calculated.

4. What is the relationship between mean and partial derivatives on a product?

The relationship between mean and partial derivatives on a product is that the mean derivative is equal to the sum of the partial derivatives of each variable in the product. This is a result of the linearity property of derivatives.

5. Are there any limitations to swapping between mean and partial derivatives on a product?

Yes, there are limitations to swapping between mean and partial derivatives on a product. This method is only applicable when the variables in the product are independent of each other. If the variables are dependent, then the mean derivative may not accurately represent the rate of change of the product.

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