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Question
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
Options
`2`
\[\frac{1}{2 \sqrt{1 - x^2}}\]
\[2/x\]
\[1 - x^2\]
Solution
`2`
\[\text { Let u } = \cos^{- 1} \left( 2 x^2 - 1 \right)\]
\[\text { Put x } = \cos\theta\]
\[ \Rightarrow \theta = \cos^{- 1} x\]
\[\frac{d\theta}{dx} = \frac{- 1}{\sqrt{1 - x^2}}\]
\[\text { Now, u } = \cos^{- 1} \left( \cos2\theta \right)\]
\[ \Rightarrow u = 2\theta\]
\[\Rightarrow \frac{du}{dx} = 2\frac{d\theta}{dx}\]
\[ \Rightarrow \frac{du}{dx} = \frac{- 2}{\sqrt{1 - x^2}} . . . \left( i \right)\]
\[\text { and,} \]
\[ v = \cos^{- 1} x\]
\[ \Rightarrow v = \cos^{- 1} \left( \cos\theta \right)\]
\[ \Rightarrow v = \theta\]
\[\frac{dv}{dx} = \frac{d\theta}{dx}\]
\[ \Rightarrow \frac{dv}{dx} = \frac{- 1}{\sqrt{1 - x^2}} . . . \left( ii \right)\]
\[\text { Dividing } \left( i \right) \text { by }\left( ii \right), \text { we get }, \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{- 2}{\sqrt{1 - x^2}} \times \frac{\sqrt{1 - x^2}}{- 1}\]
\[ \Rightarrow \frac{du}{dv} = 2\]
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