If $y=\log _2\left(\log _2 x\right)$, then $\frac{d y}{d x}=$

If $y=\log _2\left(\log _2 x\right)$, then $\frac{d y}{d x}=$
  1. $\frac{\log _2 e}{2 x \log _e x}$
  2. $\frac{1}{x \log _e x \log _e 2}$
  3. $\frac{1}{\log _e(2 x)^x}$
  4. $\frac{1}{\log _2 e \log _e x}$

Solution

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} $

Asked in: AP EAMCET 2018 (23 Apr Shift 2)

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