Differentiate the function $\log[\log(\log x^5)]$ with respect to $x$.

$\frac{5}{x \log x^5 \log(\log x^5)}$

The correct answer is Option (1) → $\frac{5}{x \log x^5 \log(\log x^5)}$ ##

Let $y = \log[\log(\log x^5)]$

On differentiating w.r.t. $x$, we get

$∴\frac{dy}{dx} = \frac{d}{dx} [\log[\log(\log x^5)]] = \frac{1}{\log(\log x^5)} \cdot \frac{d}{dx}(\log \cdot \log x^5)$

$= \frac{1}{\log(\log x^5)} \cdot \left( \frac{1}{\log x^5} \right) \cdot \frac{d}{dx} \log x^5$

$= \frac{1}{\log(\log x^5)} \cdot \frac{1}{\log x^5} \cdot \frac{d}{dx} (5 \log x) = \frac{5}{x \cdot \log(\log x^5) \cdot \log(x^5)}$