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Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 10

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Answer:  10^{\log \sin x} \times \log 10 \times \cot x

Hint: Diff by 10^{x}

Given: 10^{\log \sin x}

Solution:  Let y=10^{\log \sin x}        .........(1)

Taking log both sides

        \begin{aligned} &\log y=\log 10^{\log \sin x} \\\\ &\log y=\log \sin x \log 10 \end{aligned}
Differentiate w.r.t x,

        \begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\log 10 \frac{d}{d x} \log \sin x \\\\ &\frac{1}{y} \frac{d y}{d x}=\log 10 \frac{1}{\sin x} \frac{d}{d x} \sin x \end{aligned}

        \begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\log 10\left(\frac{1}{\sin x}\right) \cos x \\\\ &\frac{d y}{d x}=y(\log 10 \cdot \cot x) \\\\ &\frac{d y}{d x}=10^{\log \sin x} \times \log 10 \times \cot x \end{aligned}        ......[Using (1)]

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