Understanding the Derivative of x: A Scientist's Perspective

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SUMMARY

The derivative of the factorial function, denoted as x!, does not exist in the traditional sense since it is only defined for natural numbers. However, a continuous extension of the factorial is provided by the Gamma function, where x! = Gamma(x+1). The derivative of the Gamma function, represented as Gamma'(x+1), can be expressed using the digamma function, ψ(x+1), leading to the formula: Gamma'(x+1) = (x!) * ψ(x+1). Additionally, Stirling's Approximation offers a useful method for approximating factorials for large x, given by x! ≈ (x/e)^x * sqrt(2πx).

PREREQUISITES
  • Understanding of the Gamma function and its properties
  • Familiarity with the concept of derivatives in calculus
  • Knowledge of Stirling's Approximation
  • Basic understanding of the digamma function
NEXT STEPS
  • Research the Bohr–Mollerup theorem and its implications for the Gamma function
  • Learn about the properties and applications of the digamma function
  • Explore advanced topics in neutrix calculus and its applications in generalized functions
  • Study the relationship between the Gamma function and other special functions in mathematics
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced mathematical concepts related to factorials and their derivatives.

  • #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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