If \(y = \log_e\left(\cos e^{x}\right)\), where \(\cos e^{x} \gt 0\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-e^{x}\tan e^{x}\)
By the chain rule, taking log, then cosine, then \(e^{x}\):
\( \displaystyle \frac{dy}{dx} = \frac{1}{\cos e^{x}} \cdot \left(-\sin e^{x}\right) \cdot e^{x} \)
\( = -e^{x}\tan e^{x} \)