If $y=x^{{\left(\log x\right)}^{\log \left(\log x\right)}},$ then $\frac{dy}{dx}=$

$y=x^{{\left(\log x\right)}^{\log \left(\log x\right)}},$

$\log y={\left(\log x\right)}^{\log \left(\log x\right)}\times \log x$

$\text{Taking log of both sides, we get}$

$\log (\log y)=\log (\log x)+\log (\log x)\log (\log x)$

$\text{Differentiating w.r.t.}x\text{, we get}$

$\begin{array}{c}\frac{1}{\log y}\cdot \frac{1}{y}\frac{dy}{dx}=\frac{1}{x\log x}+\frac{2\log (\log x)}{\log x}\frac{1}{x}\\ =\frac{2\log (\log x)+1}{x\log x}\\ \text{or}\frac{dy}{dx}=\frac{y}{x}\cdot \frac{\log y}{\log x}(2\log (\log x)+1)\end{array}$