If \(y = (\sin x)^{x}\), where \(0 \lt x \lt \pi\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \((\sin x)^{x}\left[\log_e(\sin x) + x\cot x\right]\)
The variable is in both the base and the exponent, so we use logarithmic differentiation:
\(\log_e y = x\log_e(\sin x)\)
By the product rule:
\( \displaystyle \frac{1}{y}\frac{dy}{dx} = \log_e(\sin x) + x \cdot \frac{\cos x}{\sin x} \)
\( \displaystyle \frac{dy}{dx} = (\sin x)^{x}\left[\log_e(\sin x) + x\cot x\right] \)