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(A) Given the function $y = x \sin x$. Taking the natural logarithm on both sides: $\ln y = \ln(x \sin x)$. Using the property $\ln(ab) = \ln a + \ln

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$\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} + \cot x$

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(A) Given the function $y = x \sin x$. Taking the natural logarithm on both sides: $\ln y = \ln(x \sin x)$. Using the property $\ln(ab) = \ln a + \ln b$,we get $\ln y = \ln x + \ln(\sin x)$. Differentiating both sides with respect to $x$: $\frac{d}{dx}(\ln y) = \frac{d}{dx}(\ln x) + \frac{d}{dx}(\ln(\sin x))$. Applying the chain rule: $\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} + \frac{1}{\sin x} \cdot \cos x$. Simplifying the expression: $\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} + \cot x$.

If $y = x \sin x$,then

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