10/16/2021, 08:30 PM

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.

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.

Luca Onnis