Understanding the Derivative of x: A Scientist's Perspective

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Discussion Overview

The discussion revolves around the derivative of the factorial function, denoted as x!, and its relationship with the Gamma function. Participants explore the definitions, continuity, and differentiability of these functions, as well as various mathematical properties and approximations related to them.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that the derivative of x! does not exist because x! is only defined for natural numbers, and any differentiable function must be continuous.
  • Others propose that there is a continuous extension of the factorial function through the Gamma function, specifically noting that for integers, x! can be expressed as Gamma(x+1).
  • A participant mentions that the Gamma function is defined for all real numbers except negative integers, where it has poles, and is continuous on the positive real line.
  • Some participants discuss the relationship between the Gamma function and the factorial, asserting that while the Gamma function extends the factorial, they are not the same function and have different domains.
  • Stirling's Approximation is introduced as a way to approximate x! and is noted as a formula that can be differentiated using basic calculus tools.
  • Wolfram Alpha is referenced as providing the derivative of x! in terms of the Gamma function and the digamma function, although there is uncertainty about whether it shows the steps involved in the calculation.
  • Participants explore the implications of defining the factorial for non-integer values and the challenges that arise from this approach.
  • Some participants express uncertainty about the reliability of estimates for the derivative and discuss the behavior of the derivative as x increases.

Areas of Agreement / Disagreement

There is no consensus on the existence of the derivative of x! or the definitions and properties of the Gamma function. Multiple competing views remain regarding the continuity and differentiability of these functions, as well as their definitions and relationships.

Contextual Notes

Participants highlight limitations in definitions and continuity, particularly regarding the Gamma function's behavior at negative integers and the interpretation of factorials for non-integer values.

  • #31
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  • #33
Whenever I read an article from Wikipedia there always ends up being ten more tabs opened in my browser or something ridiculous.
 
  • #34
micromass said:
Read what exactly? What are you trying to make clear?
I think that WWGD should read the article!
because he did not know that the gamma Function can write it as X!.
this is my point.
 
  • #35
Emmanuel_Euler said:
I think that WWGD should read the article!
because he did not know that the gamma Function can write it as X!.
this is my point.

Don't worry, WWGD knows the connection between the Gamma function and the functorial pretty well. I guess you are missing his point: the Gamma function can not be written in function of the factorial. Yes, it is true that ##\Gamma(n) = (n-1)!##, but this is only for positive integers ##n##. For nonintegers, we do not have that relationship since the factorial is not defined for non-integers. By definition, the factorial is a function ##\mathbb{N}\rightarrow \mathbb{N}##. There are many extensions of this, only one of them is given by a Gamma function.
The derivative of ##n\rightarrow n!## is undefined.
 
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  • #36
Thank you for explaining.
You was right about me.
i missed his point.
 

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