The derivative of $c o s^{- 1} (2 x^{2} - 1)$ w.r.t. $c o s^{- 1} x$ is

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

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\frac{2}{x}

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2

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1 - x^{2}

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Solution

The correct option is B $2$
$y = {cos}^{- 1} (2 x^{2} - 1) & z = {cos}^{- 1} x$
$\frac{d y}{d z} = \frac{d y / d x}{d z / d x}$
$y = {cos}^{- 1} (2 x^{2} - 1)$
$\frac{d y}{d x} = \frac{- 1}{\sqrt{1 - {(2 x^{2} - 1)}^{2}}} \times [4 x - 0]$
$\frac{d y}{d x} = \frac{- 1 * 4 x}{\sqrt{1 - 4 x^{2} - 1 + 4 x^{2}}} = \frac{- 4 x}{2 x \sqrt{1 - x^{2}}}$
$\frac{d y}{d x} = \frac{- 2}{\sqrt{1 - x^{2}}} ⟶ 1$

$z = {cos}^{- 1} x$
$\frac{d z}{d x} = \frac{- 1}{\sqrt{1 - x^{2}}} ⟶ 2$
From equation 1 and 2
$\frac{d y}{d z} = \frac{- 2 / \sqrt{1 - x^{2}}}{- 1 / \sqrt{1 - x^{2}}}$
$= 2$

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