Understanding the Derivative of x: A Scientist's Perspective

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The derivative of the factorial function x! does not exist in the traditional sense, as it is only defined for natural numbers and is not continuous. However, the Gamma function serves as a continuous extension of the factorial, where x! can be expressed as Gamma(x+1). Discussions highlight that the derivative of the Gamma function, which can be computed, relates to the factorial through the digamma function. The Gamma function is continuous for positive real numbers but has poles at negative integers, complicating its behavior in that domain. Overall, while x! is not differentiable in the classical sense, the Gamma function provides a framework for exploring derivatives in a broader context.
  • #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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