The derivative of \((\log \mathrm{x})^{\mathrm{x}}\) with respect to \(\log \mathrm{x}\) is

Let \(\mathrm{u}=(\log \mathrm{x})^{\mathrm{x}}\)
\(
\begin{aligned}
& \therefore \log \mathrm{u}=\mathrm{x} \log \left[\log (\mathrm{x})^{\mathrm{x}}\right] \\
& \therefore \frac{1}{\mathrm{u}} \frac{\mathrm{du}}{\mathrm{dx}}=\frac{\mathrm{x}}{\log (\mathrm{x})} \times \frac{1}{\mathrm{x}}+\log (\log \mathrm{x})=\log (\log \mathrm{x})+\frac{1}{\log \mathrm{x}} \\
& \therefore \frac{\mathrm{du}}{\mathrm{dx}}=(\log \mathrm{x})^{\mathrm{x}}\left[\frac{1}{\log \mathrm{x}}+\log (\log \mathrm{x})\right]
\end{aligned}
\)
Let \(v=\log x \Rightarrow \frac{d v}{d x}=\frac{1}{x}\)
\(
\therefore \frac{d u}{d v}=(\log x)^x(x)\left[\frac{1}{\log x}+\log (\log x)\right]
\)