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

ConceptSimple Problems on Applications of Derivatives

#### Question

Differentiate  $\sin^{- 1} \sqrt{1 - x^2}$ with respect to $\cos^{- 1} x, \text { if}$ $x \in \left( - 1, 0 \right)$ ?

#### Solution

$\text { Let, u } = \sin^{- 1} \sqrt{1 - x^2}$
$\text { Put x} = \cos\theta$
$\Rightarrow u = \sin^{- 1} \sqrt{1 - \cos^2 \theta}$
$\Rightarrow u = \sin^{- 1} \left( \sin\theta \right) . . . \left( i \right)$
$\text { And, v} = \cos^{- 1} x . . . \left( ii \right)$
$\text { Now, x } \in \left( - 1, 0 \right)$
$\Rightarrow \cos\theta \in \left( - 1, 0 \right)$
$\Rightarrow \theta \in \left( \frac{\pi}{2}, \pi \right)$
$\text { So, from equation } \left( i \right),$
$u = \pi - \theta \left[ \text { Since }, \sin^{- 1} \left( \sin\theta \right) = \pi - \theta if \theta \in \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \right]$
$\Rightarrow u = \pi - \cos^{- 1} x \left[ \text { Since, x } = \cos\theta \right]$

Differentiating it with respect to x,

$\frac{du}{dx} = 0 - \frac{- 1}{\sqrt{1 - x^2}}$
$\Rightarrow \frac{du}{dx} = \frac{1}{\sqrt{1 - x^2}} . . . \left( iii \right)$
$\text { from equation } \left( ii \right),$
$v = \cos^{- 1} x$

Differentiating it with respect to x,

$\frac{dv}{dx} = \frac{- 1}{\sqrt{1 - x^2}} . . . \left( iv \right)$
$\text { Dividing equation } \left( iii \right) by \left( iv \right),$
$\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{1}{\sqrt{1 - x^2}} \times \frac{\sqrt{1 - x^2}}{- 1}$
$\therefore \frac{du}{dx} = - 1$

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