$\text{ If }y=a^{x^{a^{x^{.....}}}}\text{ then the value of }\frac{dy}{dx}\text{ is }$

$\begin{array}{l}y={a^x}^{{ }^{a^{x^{a^{x....}}.\dots }}}\Rightarrow y=a^{x^y}\\ \log y=x^y\log a\\ \log (\log y)=y\log x+\log (\log a)\mid \end{array}$

Diff   w.r. to ‘x’

$\begin{array}{l}\frac{1}{\log y}\cdot \frac{1}{y}\frac{dy}{dx}=\frac{y}{x}+\log x\frac{dy}{dx}+o\\ \frac{dy}{dx}\left[\frac{1}{y\log y}-\log x\right]=\frac{y}{x}\\ \frac{dy}{dx}\left[\frac{1-y\log y\log x}{y\log y}\right]=\frac{y}{x}\Rightarrow \frac{dy}{dx}=\frac{y^2\log y}{x[1-y\log y\log x]}\end{array}$