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Q.

The derivative of $\sec^{- 1} \frac{1}{2 x^{2} - 1}$ with respect to $\sqrt{1 - x^{2}}$ is

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a

$\frac{x}{2}$

b

$\frac{2}{x}$

c

0

d

x

answer is D.

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Detailed Solution

Let $y = \sec^{- 1} (\frac{1}{2 x^{2} - 1})$

Put $x = \cos θ$

$y = \sec^{- 1} (\sec 2 θ) \Rightarrow y = 2 θ$

$\frac{d y}{d θ} = 2$

$u = \sqrt{1 - x^{2}} = \sqrt{1 - \cos^{2} θ} = \sin θ$

$\frac{d u}{d θ} = \cos θ$

$\frac{d y}{d u} = \frac{2}{\cos θ} = \frac{2}{x}$

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The derivative of sec−112x2−1 with respect to 1−x2 is