Question

Differentiate given problems w.r.t.x.
$y=(logx{)}^{logx}$

Answer 1

Let $y=(logx{)}^{logx}$

Taking log on both sides, we get log $y=log\left[\right(logx{)}^{logx}]$

$\Rightarrow$ log y=log x (log x) $\because log{m}^n=nlogm$

Differentiating both sides sides w.r.t. x, we have

$\frac{1}{y}\frac{dy}{dx}=\left(logx\right)\frac{d}{dx}log\left(logx\right)+loglog\left(x\right)\frac{d}{dx}log\left(x\right)$

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

$\therefore$ $\frac{dy}{dx}=\frac{y}{x}1+log\left(logx\right)$

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