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On the first derivative of the n-th tetration of f(x) - Printable Version

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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.  Cool
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.  Cool
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.

That would be interesting , in combinations with ideas like continuum sum etc.

regards

tommy1729