Question

The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to  \[\cos^{- 1} x\]  is ___________ .

Options

• `2`

• \[\frac{1}{2 \sqrt{1 - x^2}}\]

• \[2/x\]

• \[1 - x^2\]

Answer 1

`2`

 

\[\text { Let u } = \cos^{- 1} \left( 2 x^2 - 1 \right)\]

\[\text { Put x } = \cos\theta\]

\[ \Rightarrow \theta = \cos^{- 1} x\]

\[\frac{d\theta}{dx} = \frac{- 1}{\sqrt{1 - x^2}}\]

\[\text { Now, u } = \cos^{- 1} \left( \cos2\theta \right)\]

\[ \Rightarrow u = 2\theta\]

\[\Rightarrow \frac{du}{dx} = 2\frac{d\theta}{dx}\]
\[ \Rightarrow \frac{du}{dx} = \frac{- 2}{\sqrt{1 - x^2}} . . . \left( i \right)\]
\[\text { and,} \]
\[ v = \cos^{- 1} x\]
\[ \Rightarrow v = \cos^{- 1} \left( \cos\theta \right)\]
\[ \Rightarrow v = \theta\]

\[\frac{dv}{dx} = \frac{d\theta}{dx}\]

\[ \Rightarrow \frac{dv}{dx} = \frac{- 1}{\sqrt{1 - x^2}} . . . \left( ii \right)\]

\[\text { Dividing } \left( i \right) \text { by }\left( ii \right), \text { we get }, \]

\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{- 2}{\sqrt{1 - x^2}} \times \frac{\sqrt{1 - x^2}}{- 1}\]

\[ \Rightarrow \frac{du}{dv} = 2\]