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Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 14

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

Hint: Diff by using \left(\sin ^{-1} x\right)^{x}

Given: \left(\sin ^{-1} x\right)^{x}

Solution:  Let  y=\left(\sin ^{-1} x\right)^{x}

Taking log on both sides

        \log y=x \cdot \log \left(\sin ^{-1} x\right)

Differentiate w.r.t x,

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

 Now , put the value of ,   y=\left(\sin ^{-1} x\right)^{x}

            \Rightarrow \frac{d y}{d x}=\left(\sin ^{-1} x\right)^{x}\left[\log \left(\sin ^{-1} x\right)+x \cdot \frac{1}{\sin ^{-1} x} \cdot \frac{1}{\sqrt{1-x^{2}}}\right]

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