+9519 Find the derivative of \(x^{x^{\cdot\cdot\cdot}}\)
This are infinitely many 'x's and seems impossible, but I found something really awesome.
Let \(f(x) = x^{f(x)}\) so that the solution of \(f(x)\) is \(x^{x^{\cdot\cdot\cdot}}\).
\(f(x) = x^{f(x)}\\ \ln f(x) = f(x) \cdot \ln x\\ \dfrac{1}{f(x)}\cdot\dfrac{df(x)}{dx} = \dfrac{df(x)}{dx} \cdot \ln x + \dfrac{f(x)}{x}\\ \dfrac{1}{f(x)}\cdot\dfrac{df(x)}{dx} - \dfrac{df(x)}{dx} \cdot \ln x = \dfrac{f(x)}{x}\\ \dfrac{df(x)}{dx}\left(\dfrac{1}{f(x)}-\ln x\right) = \dfrac{f(x)}{x}\\ \dfrac{df(x)}{dx} =\dfrac{f(x)}{x\left({\dfrac{1}{f(x)}-\ln x}\right)}\\ \dfrac{df(x)}{dx} = \dfrac{\left(f(x)\right)^2}{x - x\cdot f(x)\cdot \ln x}\\ \dfrac{df(x)}{dx}=\dfrac{\left(x^{x^{\cdot\cdot\cdot}}\right)^2}{x - x^{x^{x^{\cdot\cdot\cdot}}+1}\cdot \ln x}\)
