The derivative of y = ^{-1}≤ft({{1+x}-{1-x}}{2}) is

Correct Answer is : 121−x2{1}{2{1-x²}}21−x21

Correct Answer is : 121−x2{1}{2{1-x²}}21−x21

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AP EAMCET Engineering 22 Sep 2020 Shift 2 Paper

Section: Mathematics

Question : 10 of 160

Marks: +1, -0

The derivative of

y

\sin^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{2}\right)

Solution:

y=\sin^{-1}\left(\;\frac{\sqrt{1+x}-\sqrt{1-x}}{2}\right)

Put,

x=\cos 2\theta \Rightarrow 2\theta=\cos^{-1} x

\theta=\;\frac{1}{2}\cos^{-1} x

=\sin^{-1}\left(\;\frac{\sqrt{1+\cos 2\theta}-\sqrt{1-\cos 2\theta}}{2}\right)

=\sin^{-1}\left(\;\frac{\sqrt{2}\cos\theta-\sqrt{2}\sin\theta}{2}\right)

=\sin^{-1}\left(\;\frac{\cos\theta-\sin\theta}{\sqrt{2}}\right)

=\sin^{-1}\left(\;\frac{1}{\sqrt{2}}\cos\theta-\sin\theta\;\frac{1}{\sqrt{2}}\right)

=\sin^{-1}\left(\sin\frac{\pi}{4}\cos\theta-\cos\frac{\pi}{4}\sin\theta\right)

=\sin^{-1}\left[\sin\left(\;\frac{\pi}{4}-\theta\right)\right]=\;\frac{\pi}{4}-\theta

y=\;\frac{\pi}{4}-\;\frac{1}{2}\cos^{-1} x

Now, differentiate w.r.t '

x

', on both sides

\;\frac{dy}{dx}=0-\;\frac{1}{2}\cdot\;\frac{-1}{\sqrt{1-x^{2}}} \Rightarrow \;\frac{dy}{dx}=\;\frac{1}{2\sqrt{1-x^{2}}}

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