Find the derivative of the function: frac{d}{ | vedclass

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Question

(B) Let $y = (\sin x)^{\log x}$.Taking the natural logarithm on both sides:$\log y = \log x \cdot \log \sin x$.Differentiating both sides with respect

Vedclass · Accepted Answer

$(\sin x)^{\log x} \left[ \frac{1}{x} \log \sin x + \cot x \log x \right]$

Vedclass · Answer

(B) Let $y = (\sin x)^{\log x}$.Taking the natural logarithm on both sides:$\log y = \log x \cdot \log \sin x$.Differentiating both sides with respect to $x$ using the product rule:$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\log x) \cdot \log \sin x + \log x \cdot \frac{d}{dx}(\log \sin x)$.$\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} \log \sin x + \log x \cdot \frac{1}{\sin x} \cdot \cos x$.$\frac{1}{y} \frac{dy}{dx} = \frac{1}{x} \log \sin x + \cot x \log x$.Therefore,$\frac{dy}{dx} = y \left[ \frac{1}{x} \log \sin x + \cot x \log x \right]$.Substituting $y = (\sin x)^{\log x}$:$\frac{dy}{dx} = (\sin x)^{\log x} \left[ \frac{1}{x} \log \sin x + \cot x \log x \right]$.

Find the derivative of the function: $\frac{d}{dx} \{(\sin x)^{\log x}\}$

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