1. ## derivatives

Hei, guys! I wonder you can help me. here is my question: i must find derivative of x^x^x^x^x... infinitive. f(x)=(x^x^x^x...) .

2. ## Re: derivatives

Hello, thealivision!

$\text{Differentiate: }\:y \;=\;x^{x^{x^\cdots}}$

$\text{Take logs: }\:\ln(y) \;=\;\ln\left(x^{x^{x^\cdots}}\right) \quad\Rightarrow\quad \ln(y) \;=\;\underbrace{x^{x^{x^\cdots}}}}_{\text{This is }y}\ln x$

$\text{We have: }\:\ln(y) \;=\;y\ln(x)$

Differentiate implicitly: . $\frac{1}{y}y' \;=\;y\,\frac{1}{x} + y'\ln(x) \quad\Rightarrow\quad \frac{y'}{y} - y'\ln(x) \;=\;\frac{y}{x}$

Multiply by $xy\!:\;\;xy' - xyy'\ln(x) \:=\:y^2$

Factor: . $x\big[1-y\ln(x)\big]y' \;=\;y^2$

Therefore: . $y' \;=\;\frac{y^2}{x\big[1-y\ln(x)\big]}$

thanks a lot