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The derivative of $\cos^{- 1} (2 x^{2} - 1)$ with respect to $\cos^{- 1} x$ is

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$\frac{2}{x}$

$\frac{1}{2 \sqrt{1 - x^{2}}}$

$\sqrt{1 - x^{2}}$

answer is A.

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Detailed Solution

$y = \cos^{- 1} (2 x^{2} - 1)$

Put $x = \cos θ$

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

$y = 2 θ \Rightarrow \frac{d y}{d θ} = 2$

$u = \cos^{- 1} x = \cos^{- 1} (\cos θ) = θ$

$\frac{d u}{d θ} = 1$

$\frac{d y}{d u} = 2$

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