Question
Diffrentiate the following w.r.t.x
$\log _{e^2}(\log x)$
$\log _{e^2}(\log x)$
$=\frac{\log (\log x)}{2 \log e}=\frac{\log (\log x)}{2} \quad \ldots[\because \log e=1]$
Differentiating w.r.t. $x$, we get
$\begin{aligned} \frac{d y}{d x} & =\frac{1}{2} \frac{d}{d x}[\log (\log x)] \\ & =\frac{1}{2} \times \frac{1}{\log x} \cdot \frac{d}{d x}(\log x) \\ = & \frac{1}{2 \log x} \times \frac{1}{x}=\frac{1}{2 x \log x}\end{aligned}$
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