First slide

Derivatives

Question

If $y = x^{{(\log x)}^{\log (\log x)}}, t h e n \frac{d y}{d x} i s$

Moderate

A

$\frac{y}{x} (1 n x^{\log x - 1}) + 2 1 n x 1 n (1 n x)$
B

$\frac{y}{x} {(\log x)}^{\log (\log x)} (2 \log (\log x) + 1)$
C

$\frac{y}{x 1 n x} [{(1 n x)}^{2} + 21 n (1 n x)]$
D

$\frac{y}{x} \frac{\log y}{\log x} [2 \log (\log x) + 1]$

Solution

$y = x^{{(\log x)}^{\log (logx)}}$
$\Rightarrow \log y = (\log x) {(\log x)}^{\log (\log x)} \to (1)$
Taking log of both sides, we get
$\Rightarrow \log (\log y) = \log (\log x) + \log (\log x) \log (\log x)$
Diff. w.r.t x, we get
$\frac{1}{\log y} . \frac{1}{y} \frac{d y}{d x} = \frac{1}{x \log x} + \frac{2 \log (\log x)}{\log x} \frac{1}{x}$
$= \frac{2 \log (\log x) + 1}{x \log x}$
$\Rightarrow \frac{d y}{d x} = \frac{y}{x} . \frac{\log y}{\log x} (2 \log (\log x) + 1)$
Substituting the value of log y from (1), we get
$\frac{y}{x} {(\log x)}^{\log (\log x)} (2 \log (\log x) + 1)$

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