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Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 2

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Answer: x^{\sin x}\left(\cos x \ln x+\left(\frac{\sin x}{x}\right)\right)

Hint: Differentiate by x^{n}

Given: x^{\operatorname{Sin} x}

Solution: Let \mathrm{y}=x^{\operatorname{Sin} x}

Take natural log to both sides

        \begin{aligned} &\ln y=\ln x^{\sin x} \\\\ &\ln y=\sin x \ln x \end{aligned}                \left[\because \log a^{b}=b \log a\right]

Diff both side w.r.t x

        \frac{d}{d x}(\ln y)=\frac{d}{d x}(\sin x \ln x)

Using implicit diff on LHS, product rule on RHS

        \begin{aligned} &\frac{1}{y}\left(\frac{d y}{d x}\right)=\cos x \ln x+\frac{\sin x}{x} \\\\ &\frac{d y}{d x}=y\left(\cos x \ln x+\frac{\sin x}{x}\right) \end{aligned}

Substituting back for y

        \frac{d y}{d x}=x^{\sin x}\left(\cos x \ln x+\frac{\sin x}{x}\right)

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