Exponential issues 02

$\frac{\partial x^{x^{x^x}}}{\partial x} \Rightarrow x^{x^{x^x}} \left(x^{x^x-1}+x^{x^x} \log (x) \left(x^{x-1}+x^x \log (x) (\log (x)+1)\right)\right)$

Numerically searching for a root of that expression, that is, searching for a zero value for that expression, near x being 0.25, gives $x=0.274689385244017$ . Evaluating $x^{x^{x^x}}$ at $x=0.274689385244017$ gives 0.593237297769728.

Evaluating $\frac{\partial \frac{\partial x^{x^{x^x}}}{\partial x}}{\partial x}$ , $x^{x^{x^x}} \left(x^{x^x-1}+x^{x^x} \log (x) \left(x^{x-1}+x^x \log (x) (\log (x)+1)\right)\right)^2+ x^{x^{x^x}} \left(x^{x^x-1} \left(\frac{x^x-1}{x}+x^x \log (x) (\log (x)+1)\right)+ x^{x^x-1} \left(x^{x-1}+x^x \log (x) (\log (x)+1)\right)+x^{x^x} \log (x) \left(x^{x-1}+ x^x \log (x) (\log (x)+1)\right)^2+x^{x^x} \log (x) \left(x^{x-1} \log (x)+ x^{x-1} (\log (x)+1)+x^{x-1} \left(\frac{x-1}{x}+\log (x)\right)+x^x \log (x) (\log (x)+1)^2\right)\right)$ , at $x=0.274689385244017$ gives 3.66815838086362, which is a positive number and therefore the root is a minimum.