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Question
Differentiate \[x^{\cos^{- 1} x}\] ?
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Solution
\[\text{ Let y } = x^{\cos^{- 1} x} . . . \left( i \right)\]
Taking log both sides
\[\log y = \log x^{\cos^{- 1} x} \]
\[ \Rightarrow \log y = \cos^{- 1} x \log x\]
Differentiating with respect to x,
\[\frac{1}{y}\frac{dy}{dx} = \cos^{- 1} x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx} \cos^{- 1} x \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \cos^{- 1} x\left( \frac{1}{x} \right) + \log x\left( \frac{- 1}{\sqrt{1 - x^2}} \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{\cos^{- 1} x}{x} - \frac{\log x}{\sqrt{1 - x^2}}\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{\cos^{- 1} x}{x} - \frac{\log x}{\sqrt{1 - x^2}} \right]\]
\[ \Rightarrow \frac{dy}{dx} = x^{\cos^{- 1} x} \left[ \frac{\cos^{- 1} x}{x} - \frac{\log x}{\sqrt{1 - x^2}} \right] \left[ \text{ Using equation} \left( i \right) \right]\]
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