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# Solution for Differentiate Tan − 1 ( 2 X 1 − X 2 ) with Respect to Cos − 1 ( 1 − X 2 1 + X 2 ) , If 0 < X < 1 ? - CBSE (Science) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

Differentiate $\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)$ with respect to $\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1$ ?

#### Solution

$\text { Let, u } = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)$

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

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

$\Rightarrow u = \tan^{- 1} \left( \tan2\theta \right) . . . \left( i \right)$

$\text { let, v} = \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right)$

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

$\Rightarrow v = \cos^{- 1} \left( \cos2\theta \right) . . . \left( ii \right)$

$\text { Here, } 0 < x < 1$

$\Rightarrow 0 < \tan\theta < 1$

$\Rightarrow 0 < \theta < \frac{\pi}{4}$

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

$u = 2\theta \left[ \text { Since }, \tan^{- 1} \left( \tan\theta \right) = \theta,\text { if } \theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \right]$

$\Rightarrow u = 2 \tan^{- 1} x \left[ \text { Since,} x = \tan\theta \right]$

differentiating it with respect to x,

$\frac{du}{dx} = \frac{2}{1 + x^2} . . . \left( iii \right)$

$\text { From equation } \left( ii \right),$

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

$\Rightarrow v = 2 \tan^{- 1} x \left[ \text { Since, } x = \tan\theta \right]$

Differentiating it with respect to x,

$\frac{dv}{dx} = \frac{2}{1 + x^2} . . . \left( iv \right)$

$\text { Dividing equation } \left( iii \right) \text { by }\left( iv \right),$

$\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{2}{1 + x^2} \times \frac{1 + x^2}{2}$

$\therefore \frac{du}{dv} = 1$

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Solution Differentiate Tan − 1 ( 2 X 1 − X 2 ) with Respect to Cos − 1 ( 1 − X 2 1 + X 2 ) , If 0 < X < 1 ? Concept: Simple Problems on Applications of Derivatives.
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