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Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 7

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Answer: (\sin )^{\cos x}(\cos x \cot x-\sin \log (\sin x)


Given: (\sin x)^{\cos x}

Solution:  Let y=(\sin x)^{\cos x}

Taking log both sides

        \log y=\log (\sin x)^{\cos x}                  \left[\because \log m^{n}=n \log m\right] 

        \log y=\cos x \cdot \log (\sin x)
Differentiate w.r.t x,

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

        \frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{1}{\sin x} \cdot \cos x+\log (\sin x)(-\sin x)

        \begin{aligned} &\frac{d y}{d x}=y(\cos x \cdot \cot x-\sin (\log (\sin x)) \\\\ &(\sin x)^{\cos x}(\cos x \cdot \cot x-\sin \log (\sin x)) \end{aligned}

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