If \(y = \log_e(\tan x)\), \(0 \lt x \lt \displaystyle\frac{\pi}{2}\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\sec x\,\mathrm{cosec}\,x\)
Since \(\frac{d}{du}\log_e u = \frac{1}{u}\), the chain rule gives
\( \displaystyle \frac{dy}{dx} = \frac{1}{\tan x} \cdot \sec^{2} x \)
Writing in terms of sine and cosine:
\( \displaystyle \frac{dy}{dx} = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^{2} x} \)
\( \displaystyle = \frac{1}{\sin x\cos x} \)
\( = \sec x\,\mathrm{cosec}\,x \)