Differentiate the function log_7(log x) with respect to x.

The correct answer is Option (4) → $\frac{1}{x \log x \ln 7}$ ##

Let $y = \log_7(\log x) = \frac{\log(\log x)}{\log 7}$ (by change of base formula).

The function is defined for all real numbers $x > 1$. Therefore

$\frac{dy}{dx} = \frac{1}{\log 7} \frac{d}{dx}(\log(\log x))$

$= \frac{1}{\log 7} \frac{1}{\log x} \cdot \frac{d}{dx}(\log x)$

$= \frac{1}{x \log 7 \log x}$