# Tetration Forum

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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.
Cool.

Not sure if this is new.

But somehow this must have applications ...

regards

tommy1729
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
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
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.
(10/27/2021, 03:35 PM)Luknik Wrote: [ -> ]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.

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

regards

tommy1729