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Solution for Differentiate ( Sin X ) Cos X ? - CBSE (Science) Class 12 - Mathematics

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Differentiate \[\left( \sin x \right)^{\cos x}\] ?


\[\text{ Let y} = \left( \sin x \right)^{\cos x }. . . \left( i \right)\]
\[\text{ Taking \log on both sides}, \]
\[\log y = \log \left( \sin x \right)^{\cos x }\]
\[ \Rightarrow \log y = \cos x \log \sin x \]
\[\text{ Differentiating with respect to x }, \]
\[\frac{1}{y}\frac{dy}{dx} = \cos x\frac{d}{dx}\left( \log \sin x \right) + \log \sin x\frac{d}{dx}\left( \cos x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \cos x\frac{1}{\sin x}\frac{d}{dx}\left( \sin x \right) + \log \sin x\left( - \sin x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{\cos x}{\sin x}\left( \cos x \right) - \sin x \log \sin x\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \cos x \cot x - \sin x \log \sin x \right]\]
\[ \Rightarrow \frac{dy}{dx} = \left( \sin x \right)^{\cos x} \left[ \cos x \cot x - \sin x \log \sin x \right] \left[ \text{using equation }\left( i \right) \right]\]

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Solution for question: Differentiate ( Sin X ) Cos X ? concept: Simple Problems on Applications of Derivatives. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
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