Limit Continuity And Differentiability Ques 17
- The derivative of $\sec ^{-1}\left(-\frac{1}{2 x^2-1}\right)$ with respect to $\sqrt{1-x^2}$ at $x=\frac{1}{2}$ is
(1986, 2M)
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Answer:
Correct Answer: 17.$(-4)$
Solution: Let $u=\sec ^{-1}\left(-\frac{1}{2 x^2-1}\right)$ and $v=\sqrt{1-x^2}$
Put $ x=\cos \theta $
$\therefore \quad$ $u=\sec ^{-1}(-\sec 2 \theta)$ $\theta)$ and $v=\sin \theta$
$\Rightarrow \quad$ $u=\pi-2 \theta$
$\left[\because \quad\sec ^{-1}(-x)=\pi-\sec ^{-1} x\right]$
and $v=\sin \theta$
$\Rightarrow \quad \frac{d u}{d \theta}=-2$
and $\frac{d v}{d \theta}=\cos \theta$
$\Rightarrow \quad \frac{d u}{d v}=-\frac{2}{\cos \theta},\left(\frac{d u}{d v}\right)_{\theta-\pi / 3}=-4$
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