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# Solution for Differentiate Cos − 1 ( 1 − X 2 N 1 + X 2 N ) , < X < ∞ ? - CBSE (Science) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

Differentiate $\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty$ ?

#### Solution

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

$\text{ Put } x^n = \tan\theta$

$\therefore y = \cos^{- 1} \left[ \frac{1 - \left( x^n \right)^2}{1 + \left( x^n \right)^2} \right]$

$\Rightarrow y = \cos^{- 1} \left( \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \right)$

$\Rightarrow y = \cos^{- 1} \left( \cos 2\theta \right) . . . \left( i \right)$

$\text{ Here }, 0 < x < \infty$

$\Rightarrow 0 < x^n < \infty$

$\Rightarrow 0 < \tan\theta < \infty$

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

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

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

$y = 2\theta \left[ \text{ Since }, \cos^{- 1} \left( \cos \theta \right) = \theta, if \theta \in \left[ 0, \pi \right] \right]$

$\Rightarrow y = 2 \tan^{- 1} \left( x^n \right)$

Differentiate it with respect to x using chain rule,

$\frac{d y}{d x} = 2\left[ \frac{1}{1 + \left( x^n \right)^2} \right]\frac{d}{dx}\left( x^n \right)$

$\Rightarrow \frac{d y}{d x} = \frac{2}{1 + x^{2n}} \times \left( n x^{n - 1} \right)$

$\therefore \frac{d y}{d x} = \frac{2n x^{n - 1}}{1 + x^{2n}}$

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Solution Differentiate Cos − 1 ( 1 − X 2 N 1 + X 2 N ) , < X < ∞ ? Concept: Simple Problems on Applications of Derivatives.
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