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If $y=\log _2\left(\log _2 x\right)$, then $\frac{d y}{d x}=$
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Verified Answer
The correct answer is:
$\frac{1}{x \log _e x \log _e 2}$
Given, $y=\log _2\left(\log _2 x\right) \Rightarrow y=\frac{\log _e\left(\frac{\log _e x}{\log _e 2}\right)}{\log _{e^2}}$ On differentiating w.r.t. to ' $x$ ', we are getting
$$
\begin{aligned}
\frac{d y}{d x} & =\frac{1}{\log _e 2 \frac{\log _e x}{\log _e 2}} \times \frac{1}{x \cdot \log _e 2} \\
\Rightarrow \quad \frac{d y}{d x} & =\frac{1}{x \cdot \log _e x \log _e 2} .
\end{aligned}
$$
$$
\begin{aligned}
\frac{d y}{d x} & =\frac{1}{\log _e 2 \frac{\log _e x}{\log _e 2}} \times \frac{1}{x \cdot \log _e 2} \\
\Rightarrow \quad \frac{d y}{d x} & =\frac{1}{x \cdot \log _e x \log _e 2} .
\end{aligned}
$$
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