Find d y d x , if y=( x)^x - Vidyadip

Question

Find $\frac{d y}{d x}$, if $y=(\log x)^x$

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Answer

$y=(\log x)^x$
$\log y=\log (\log x)^x$
$\log y=x \log (\log x)$
Differentiate $w.r.t$ to $ x$
$\frac{1}{y} \frac{d y}{d x}=x \times \frac{1}{\log x} \times \frac{1}{x}+\log \times 1$
$\frac{1}{y} \frac{d y}{d x}=\frac{1}{\log x}+\log (\log x)$
$\frac{d y}{d x}=y\left(\frac{1}{\log x}+\log (\log x)\right)$

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