General formula of first order derivative of the nth tetration of ln(x). [closed]

A follow-up question to the previous question.

Is there a general formula of the first order derivative of

$$\ln(x)↑↑n$$

?

Where the $n$ is a constant independent of $x$.

closed as off-topic by user223391, Leucippus, JonMark Perry, Rohan, Claude LeiboviciDec 31 '17 at 10:19

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• There is no question here – user223391 Dec 30 '17 at 16:58
• Can you please write a more understandable question? – user507623 Dec 30 '17 at 16:58
• I think he means: $y = \ln (x)$ and we have to find $\frac d{dx} y^y^y^y^y^...y$. $n$ times. – Mohammad Zuhair Khan Dec 30 '17 at 17:01
• @MohammadZuhairKhan yes that is correct – Fred Wieser Dec 30 '17 at 17:02
• sorry for the confusion – Fred Wieser Dec 30 '17 at 17:02

Consider the function $f(x,y) = y^x$. If one considers $x, y$ to be functions of some variable $t$, then the multivariate chain rules says \begin{align}\frac{df}{dt} &= \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\\&=(y^x\ln y)x' + (xy^{x-1})y'\\&=f(x,y)\left(x'\ln y + y'\frac xy\right)\end{align}
Now consider $g_n(x) = x↑↑n$ and $h_n(x) = \ln g_n(x)$. Clearly $g_n(x) = f(x,g_{n-1}(x))$, so $$\frac{dg_n}{dx} = g_n(x)\left(1\cdot\ln g_{n-1}(x) + \frac{dg_{n-1}}{dx}\frac{x}{g_{n-1}(x)}\right)$$ I.e., $$\frac{d(\ln g_n)}{dx} = \ln g_{n-1}(x) + x\frac{d(\ln g_{n-1})}{dx}\\h_n' = h_{n-1} + xh_{n-1}'$$
This recursion formula can be used to find the logarithmic derivative of $g_n(x) = x↑↑n$ for any $n$. And the normal derivative is obtained by multiplying by $x↑↑n$.
Finally, the derivative you are asking for is of $g_n(\ln x)$, which can be obtained by the chain rule.