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Solution for Differentiate Cos − 1 { √ 1 + X 2 } , − 1 < X < 1 ? - CBSE (Science) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

Question

Differentiate $\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1$ ?

Solution

$\text{ Let, y }= \cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}$

$\text{Put x } = \cos2\theta$

$y = \cos^{- 1} \left\{ \sqrt{\frac{1 + \cos 2\theta}{2}} \right\}$

$y = \cos^{- 1} \left\{ \sqrt{\frac{2 \cos^2 \theta}{2}} \right\}$

$y = \cos^{- 1} \left( \cos\theta \right) . . . \left( i \right)$

$\text{ Here }, - 1 < x < 1$

$\Rightarrow - 1 < \cos2\theta < 1$

$\Rightarrow 0 < 2\theta < \pi$

$\Rightarrow 0 < \theta < \frac{\pi}{2}$

$\text{ So, from equation } \left( i \right)$

$y = \theta \left[ \text{ since } , \cos^{- 1} \left( cos\theta \right) = \theta, if\theta \in \left[ 0, \pi \right] \right]$

$\Rightarrow y = \frac{1}{2} \cos^{- 1} x \left[ \text{ Since } , x = \cos2\theta \right]$

$\text{ Differentiating it with respect to x },$

$\frac{d y}{d x} = - \frac{1}{2\sqrt{1 - x^2}}$

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Solution Differentiate Cos − 1 { √ 1 + X 2 } , − 1 < X < 1 ? Concept: Simple Problems on Applications of Derivatives.
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