If $y =log_2(log_2x),$ then $\frac{dy}{dx}$ is equal to :

$\frac{log_2e}{xlog_ex}$

The correct answer is Option (3) → $\frac{log_2e}{xlog_ex}$ ##

$y = \log_2(\log_2 x)$

Step 1: Convert using change of base

$y = \frac{\ln(\log_2 x)}{\ln 2}$

Step 2: Differentiate

$\frac{dy}{dx} = \frac{1}{\ln 2} \cdot \frac{1}{\log_2 x} \cdot \frac{d}{dx}(\log_2 x)$

Step 3: Differentiate inner term

$\frac{d}{dx}(\log_2 x) = \frac{1}{x \ln 2}$

Step 4: Substitute

$\begin{aligned} \frac{dy}{dx} &= \frac{1}{\ln 2} \cdot \frac{1}{\log_2 x} \cdot \frac{1}{x \ln 2} \\ &= \frac{1}{x (\ln 2)^2 \log_2 x} \end{aligned}$

Step 5: Simplify

Using the identities $\frac{1}{\ln 2} = \log_2 e$ and $(\log_2 x)(\ln 2) = \ln x$:

$\Rightarrow \frac{dy}{dx} = \frac{\log_2 e}{x \ln x}$