If \(y = \sin(\log_e x)\), \(x \gt 0\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{\cos(\log_e x)}{x}\)
By the chain rule, with \(u = \log_e x\) and \(\frac{du}{dx} = \frac{1}{x}\):
\( \displaystyle \frac{dy}{dx} = \cos(\log_e x) \cdot \frac{1}{x} \)
\( \displaystyle = \frac{\cos(\log_e x)}{x} \)