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Question
Differentiate \[x^{\sin x}\] ?
Solution
\[\text{ Let y }= x^{\sin x} . . . \left( i \right)\]
Taking log on both sides,
\[\log y = \log x^{\sin x} \]
\[ \Rightarrow \log y = \sin x \log x \left[ \because \log a^b = b \log a \right]\]
Differentiating with respect to x, we get,
\[\Rightarrow \frac{1}{y}\frac{dy}{dx} = \sin x\frac{d}{dx}\log x + \log x\frac{d}{dx}\sin x \left[ \text{ using product rule } \right]\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \sin x\left( \frac{1}{x} \right) + \log x\left( \cos x \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{\sin x}{x} + \left( \log x \right)\left( \cos x \right) \right]\]
\[\Rightarrow \frac{dy}{dx} = x^{\sin x} \left[ \frac{\sin x}{x} + \left( \log x \right)\left( \cos x \right) \right]\] [From (i)]
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