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On the first derivative of the n-th tetration of f(x) - Luknik - 10/16/2021
Hello! I would like to share with you my formula to compute the first derivative of the \( n\)-th tetration of \(f(x) \). In particular we have the following theorem: Let \(f(x) \) be a differentiable function and \(n \in\mathbb{N} \) , \(n\geq 2 \). Hence: \( \frac{d}{dx}{^{n}f(x)}={^{n}f(x)}{^{n-1}f(x)}\frac{f'(x)}{f(x)}\Bigl\{\sum_{j=0}^{n-2}\Bigl\{\Bigl[\prod_{j}^{n-2}{^{j}f(x)}\Bigr]\Bigl[\log\Bigl(f(x)\Bigr)\Bigr]^{n-j-1}\Bigl\}+1\Bigr\} \) For example 1 consider \(g(x)=x^{x^{x^{x}}}={^{4}x} \). \( g'(x)=x^{x^{x^{x}}}x^{x^{x}}\frac{1}{x}\Bigl\{x^{x}x\Bigl[[\log(x)]^3+[\log(x)]^2\Bigr]+x^{x}\log(x)+1\Bigr\} \) For example 2 consider \(h(x)=(\sin x)^{(\sin x)^{(\sin x)}}={^{3}\sin(x)} \). \( h'(x)=(\sin x)^{(\sin x)^{(\sin x)}}(\sin x)^{(\sin x)}(\cot x)\Bigl\{(\sin x)[(\log(\sin x))^2+\log(\sin x)]+1\Bigr\} \) I prove this by induction on \( n\) , here below you can download the paper. Have a good day! Thank you for your attention. RE: On the first derivative of the n-th tetration of f(x) - tommy1729 - 10/26/2021
Cool. Not sure if this is new. But somehow this must have applications ... regards tommy1729 RE: On the first derivative of the n-th tetration of f(x) - Luknik - 10/27/2021
I don't know! I didn't find anything like this online..I only found some examples for some specific cases but I have never found a generalization , but maybe I'm wrong. Anyway, maybe it could be possible to have a formula to compute the m-th derivative of the n-th tetration of a function, although it'll be very complicated to use. Who knows! Regards, Luknik RE: On the first derivative of the n-th tetration of f(x) - Daniel - 10/27/2021
Howdy, You might be interested in my combinatoric solution to the m^th derivative of f^n(z). See https://www.tetration.org/Combinatorics/index.html . Daniel RE: On the first derivative of the n-th tetration of f(x) - Luknik - 10/27/2021
Thank you Daniel! I'll check it out. I really hope that a closed formula for the m-th derivative of the n-th tetration of some functions exists. For example for \( f(x)=x \). So..if \( g(x)={^{n}x} \) Maybe we can find a closed formula for: \( \frac{d^{m}}{dx^{m}} {^{n}x} \) where \( n,m\in\mathbb{N} \) and \( n\geq 2 \) I have already found somethig interesting, I'll post some of the results I got. Thank you, regards. RE: On the first derivative of the n-th tetration of f(x) - tommy1729 - 10/28/2021
(10/27/2021, 03:35 PM)Luknik Wrote: Thank you Daniel! I'll check it out. That would be interesting , in combinations with ideas like continuum sum etc. regards tommy1729 |