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Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 35

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Answer:  \frac{d y}{d x}=\cos \left(x^{x}\right) \cdot x^{x}(1+\log x)

Hint: Differentiate the equation by taking log on both the sides

Given: y=\sin (x)^{x}


Let take u=(x)^{x}

        \begin{aligned} &y=\sin u \\\\ &\frac{d y}{d x}=\cos u \cdot \frac{d u}{d x} \end{aligned}

        \frac{d}{d x}(\log u)=\frac{d}{d x}(x \log x)

        \begin{aligned} &\frac{1}{u} \cdot \frac{d u}{d x}=\log x \cdot \frac{d x}{d x}+x \cdot \frac{d}{d x}(\log x) \\\\ &\frac{d u}{d x}=u\left[\log x+x \cdot \frac{1}{x}\right] \\\\ &\frac{d u}{d x}=x^{x}[\log x+1] \end{aligned}

Put the value of  du/dx we get

        \frac{d y}{d x}=\cos \left(x^{x}\right) \cdot x^{x} \cdot \log (x+1)


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