Differentiation of log[log(log x^5)] with respect to x is

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

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

Differentiate using chain rule:

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

$\frac{dy}{dx} = \frac{1}{\log(\log(x^5))} \cdot \frac{1}{\log(x^5)} \cdot \frac{1}{x^5} \cdot 5x^4$

$ \frac{dy}{dx} = \frac{5x^4}{x^5 \log(x^5) \log(\log(x^5))} = \frac{5}{x \log(x^5) \log(\log(x^5))}$