Question

Differentiate the following functions with respect to $x$
$(\log x{)}^{\log x}$

Answer 1

Let $y=(\log x{)}^{\log x}$

$\Rightarrow y=e^{\left(\log x\right)\log \left(\log x\right)}$ [ $\because y=a^{x}canbewrittenasy=e^{x\log a}$]

$\Rightarrow \frac{dy}{dx}=e^{\left(\log x\right)\log \left(\log x\right)}\frac{d}{dx}\left\{\right(\log x\left)\log \right(\log x\left)\right\}$

$=(\log x{)}^{\log x}\left\{\frac{d}{dx}\right(\log x).\log (\log x)+\log x\frac{d}{dx}\log (\log x\left)\right\}$

$=(\log x{)}^{\log x}\{\frac{\log \left(\log x\right)}{x}+\log x\times \frac{1}{\log x}\times \frac{1}{x}\}$

$=(\log x{)}^{\log x}\{\frac{\log \left(\log x\right)}{x}+\frac{1}{x}\}=(\log x{)}^{\log x}\left\{\frac{1+\log \left(\log x\right)}{x}\right\}$