Assertion (A) $\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}$ $\left(\cot…

Assertion (A) $\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)=\frac{x^2 \sin x}{\log x}$ $\left(\cot x+\frac{2}{x}-\frac{1}{x \log x}\right)$ $\operatorname{Reason}(\mathbf{R}) \frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u v}{w}\left[\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}+\frac{w^{\prime}}{w}\right]$
  1. $A$ is true, $R$ is true and $R$ is correct explanation of A
  2. $A$ is true, $R$ is true and $R$ is not correct explanation of $A$
  3. $A$ is true, $R$ is not correct
  4. $A$ is not correct, $R$ is correct

Solution

Assertion : $\frac{d}{d x}\left(\frac{x^2 \sin x}{\log x}\right)$ $\begin{aligned} & =\frac{(\log x)\left[x^2 \cos x+2 x \sin x\right]-x \sin x}{(\log x)^2} \\ & =\frac{x^2 \cos x \log x+2 x \sin x \log x-x \sin x}{(\log x)^2}\end{aligned}$ $=\frac{x^2 \sin x}{\log x}\left[\cot x+\frac{2}{x}-\frac{1}{x \log x}\right]$ Reason : $\frac{d}{d x}\left(\frac{u v}{w}\right)=\frac{u w}{w}\left[\frac{u^{\prime}}{u}+\frac{v^{\prime}}{v}+\frac{w^{\prime}}{w}\right]$

Asked in: AP EAMCET 2022 (05 Jul Shift 1)

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