If \(y = x^{\sin x}\), then value of \(x^{-\sin x}\displaystyle\frac{dy}{dx} - \cos x\log x\) is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{\sin x}{x}\)
The variable appears in both the base and the power, so logarithmic differentiation is used:
\(\log y = \sin x\,\log x\)
Differentiating both sides with respect to x (product rule on the right):
\( \displaystyle \frac{1}{y}\frac{dy}{dx} = \cos x\,\log x + \frac{\sin x}{x} \)
Since \(y = x^{\sin x}\), the left side is \(x^{-\sin x}\displaystyle\frac{dy}{dx}\). So
\( \displaystyle x^{-\sin x}\frac{dy}{dx} - \cos x\log x = \frac{\sin x}{x} \)