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  • Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 15

Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 15

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Answer:    x^{\sin ^{-1} x}\left[\frac{\sin ^{-1} x}{x}+\frac{\log x}{\sqrt{1-x^{2}}}\right]

Hint: Diff by x^{\sin ^{-1} x}

Given:x^{\sin ^{-1} x}

Solution:  Let y=x^{\sin ^{-1} x}

Taking log on both sides

        \begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\sin ^{-1} x \cdot \log x \\\\ &\frac{1}{y} \frac{d y}{d x}=\frac{\log x}{\sqrt{1-x^{2}}}+\frac{\sin ^{-1} x}{x} \end{aligned}

        \begin{aligned} &\frac{d y}{d x}=y\left[\frac{\log x}{\sqrt{1-x^{2}}}+\frac{\sin ^{-1} x}{x}\right] \\\\ &\frac{d y}{d x}=x^{\sin ^{-1} x}\left[\frac{\sin ^{-1} x}{x}+\frac{\log x}{\sqrt{1-x^{2}}}\right] \end{aligned}

 

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