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Explain solution RD Sharma class 12 chapter Differentiation exercise 10.5 question 4 maths

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Answer: x^{\cos ^{-1} x}\left(\frac{\cos ^{-1} x}{x}-\frac{\log x}{\sqrt{1-x^{2}}}\right)

Hint: Differentiate by function x^{n}x

Given:  x^{\cos ^{-1} x}

Solution:  Let y=x^{\cos ^{-1} x}

y=e^{\cos ^{-1} x \log x}                        \left[\because a^{b}=e^{b \log x}\right]

Differentiate w.r.t x

        e^{\cos ^{-1} x \log x}\left(\frac{d}{d x}\right) \cos ^{-1} \log x

        x^{\cos ^{-1} x}\left(\cos ^{-1} x\left(\frac{d}{d x}\right) \log x+\log x \frac{d}{d x} \cos ^{-1} x\right)

        x^{\cos ^{-1} x}\left[\cos ^{-1} x \cdot \frac{1}{x}+\log x \cdot\left(-\frac{1}{\sqrt{1-x^{2}}}\right)\right]

        x^{\cos ^{-1} x}\left[\frac{\cos ^{-1} x}{x}-\frac{\log x}{\sqrt{1-x^{2}}}\right]

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