Approximate derivative of x!
Working experimentally, I found the approach
[TEX](x!)'=x!(\frac{x2}{2(x1)^2}+lnx)[/TEX] For x>2 the approach improves as the value of x increases. I don't know if this is true for very large values of x, (x € R). Can you check it? 
[QUOTE=othermath;545004]I don't know if this is true for very large values of x, (x € R).
[/QUOTE] What is x! for x € R ? Define it and then we can talk about derivatives. 
[QUOTE=Batalov;545006]What is x! for x € R ?
Define it and then we can talk about derivatives.[/QUOTE] I want to say that x is a real number. 
And what do you want to say the factorial of a real number is?

Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?

[QUOTE=retina;545011]Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?[/QUOTE]
You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful [URL="https://en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plus_sin_pi_z.svg"]to demonstrate that both are valid analytic continuations of the factorials to the nonintegers[/URL]. Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives? 
[QUOTE=VBCurtis;545010]And what do you want to say the factorial of a real number is?[/QUOTE]
I assume ... 
[QUOTE=Batalov;545012]You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful [URL="https://en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plus_sin_pi_z.svg"]to demonstrate that both are valid analytic continuations of the factorials to the nonintegers[/URL].
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?[/QUOTE] Γ(n+1) requires the use of an integral. My relationship is simpler. 
[QUOTE=retina;545011]Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?[/QUOTE]
That I answered Batalov. 
[QUOTE=Batalov;545012]You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful [URL="https://en.wikipedia.org/wiki/Gamma_function#/media/File:Gamma_plus_sin_pi_z.svg"]to demonstrate that both are valid analytic continuations of the factorials to the nonintegers[/URL].
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?[/QUOTE]Thanks. Makes sense. 
There is a fine little book (good luck finding a copy!) entitled [u]The Gamma Function[/u] by Emil Artin. In it he shows that the Gamma function is distinguished by being "log convex."
As [b]Retina[/b] has noted, x! = Γ(x+1) when x is a nonnegative integer. As to the derivative: There is a wellknown asymptotic expansion [Stirling's asymptotic series] for ln(Γ(z)), z a complex variable. Taking the derivative term by term gives an asymptotic series for Γ'(z)/Γ(z). 
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