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Question

Mathematics Question on Differentiability

The derivative of sin⁡−1x\sin^{-1} x w.r.t cos⁡−11−x2\cos^{- 1} \sqrt{1-x^{2}} is

A

11−x2\frac{1}{\sqrt{ 1 -x^2}}

B

1

C

cos⁡−1x\cos^{-1} x

D

tan⁡−1x1−x2\tan^{-1} \frac{x}{ \sqrt{ 1 - x^2}}

Answer

1

Explanation

Solution

Let y=sin⁡−1xy =\sin^{-1}x and z=cos⁡−1(1−x2)z = \cos^{-1}\left(\sqrt{1-x^{2}}\right) =cos⁡−1(1−sin⁡2y) = \cos^{-1} \left(\sqrt{1-\sin^{2}y}\right) =cos⁡−1(cos⁡y)=y = \cos^{-1}\left(\cos y \right) = y Now dydz=1 \frac{dy}{dz} = 1. Hence reqd. derivative = 1.