the most fun derivative of x^x^x^...
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a reference solution for the derivative of the infinite tetration, both to credit the method and to check their own answers against it.
Recommended by 0 people across 0 different months, 2018 to 2020.
What it is
A short blackpenredpen video showing how to differentiate the infinite power tower x^x^x^... by setting y = x^y, taking logs and using implicit differentiation. It is presented as a worked example on a whiteboard-style setup.
What people said
Cites Suchetan Dontha · 2018-04 · Mathematics Stack Exchange · on “What is the antiderivative of $x^{x^{x^{x...}}}$?”
@Szeto I just found the derivative and replaced the x terms in terms of u. I verified my answer with this video: [this video]
Cites John Preston · 2019-08 · Mathematics Stack Exchange · on “Proving the derivation of the infinite tetration via Summation/Product notation is equivalent to the derivation via Implicit Differentiation”
First some background info. Feel free to skip to the bold text below if you already know this. I) The derivative of the infinite tetration of $x$ (I guess that's how you say it) or ${}^\infty x$ can be found using the following method: $$y=x\color{red}{{}^{x^{x^{.^{.^.}}}}},\quad y=\color{red}{x^{x^{x^{.^{.^.}}}}} \therefore \quad y=x^y$$ Thanks blackpenredpen [this video] ! Now take the natural log of both sides and implicity derive the equation: $$\frac{\mathrm d}{\mathrm dx}\left(\ln(y)=y\cdot \ln(x)\right) \Rightarrow \frac{1}{y}\frac{\mathrm dy}{\mathrm dx} = \frac{y}{x} + \ln x \frac{\mathrm dy}{\mathrm dx}$$ Then solve for $\frac{\mathrm dy}{\mathrm dx}$ to get the following equation: $$\f
Cites PrincessEev · 2020-12 · Mathematics Stack Exchange · on “Solution verification: Derivative of the infinite power tower $y(x) = x^{x^⋰}$”
I can't really see any errors with this myself, though I do notice your solution is different from the solution blackpenredpen derived on his channel here [this video] , which makes me wonder if there's a reason for the difference.