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Please solve RD Sharma class 12 chapter Differentiation exercise 10.2 question 56 maths textbook solution

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Answer: \frac{-2 \log x \sin (\log x)^{2}}{x}

Hint: you must know the rule of solving derivative of logarithm and trigonometric functions

Given: \cos (\log x)^{2}


Let  y=\cos (\log x)^{2}

Differentiate with respect to x

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

\begin{aligned} &\frac{d y}{d x}=-\sin (\log x)^{2} \cdot 2 \log x \frac{d}{d x} \log x \\\\ &\frac{d y}{d x}=-\sin (\log x)^{2} \cdot \frac{2 \log x}{x} \\\\ &\frac{d y}{d x}=\frac{-2 \log x \sin (\log x)^{2}}{x} \end{aligned}

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