Q106 / 160 · AP EAMCET 2018 (23 Apr Shift 2) · Mathematics

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

  • A
    $\frac{\log _2 e}{2 x \log _e x}$
  • Correct answer
    B
    $\frac{1}{x \log _e x \log _e 2}$
  • C
    $\frac{1}{\log _e(2 x)^x}$
  • D
    $\frac{1}{\log _2 e \log _e x}$

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 $$ \begin{aligned} \frac{d y}{d x} & =\frac{1}{\log _e 2 \frac{\log _e x}{\log _e 2}} \times \frac{1}{x \cdot \log _e 2} \\ \Rightarrow \quad \frac{d y}{d x} & =\frac{1}{x \cdot \log _e x \log _e 2} . \end{aligned} $$

Hence, the correct answer is option: 'B'.

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