If $y=\sin ^{-1}\left(\frac{\log x^2}{1+(\log x)^2}\right)$, then $\left(\frac{\mathrm{d} y}{\mathrm{~d}…

If $y=\sin ^{-1}\left(\frac{\log x^2}{1+(\log x)^2}\right)$, then $\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{\mathrm{d} x-1}=$
  1. 2
  2. $\frac{1}{2}$
  3. $\cdot \frac{2}{3}$
  4. -2

Solution

$y=\sin ^{-1}\left(\frac{\log x^2}{1+(\log x)^2}\right)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right)$
Let $\log x=\tan \theta$ $\begin{array}{ll} \therefore & y=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan 2 \theta}\right)=\sin ^{-1}(\sin 2 \theta)=2 \theta \\ \therefore & y=2 \tan ^{-1}(\log x) \\ \therefore & \frac{\mathrm{d} y}{\mathrm{~d} x}=2 \times \frac{1}{1+(\log x)^2} \times \frac{1}{x} \\ \therefore & \left.\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)\right|_{\mathrm{at} x=1}=2 \end{array}$

Asked in: MHT CET 2024 (11 May Shift 1)

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