If \(y = \log_e(\log_e x)\), \(x \gt 1\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\frac{1}{x\log_e x}\)
Let \(u = \log_e x\), so \(y = \log_e u\). By the chain rule:
\( \displaystyle \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} \)
\( \displaystyle = \frac{1}{\log_e x} \cdot \frac{1}{x} \)
\( \displaystyle = \frac{1}{x\log_e x} \)