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Question
Find `dy/dx` in the following:
`y = sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x < 1/sqrt2`
Solution
y = `sin^-1 (2x sqrt(1 - x^2))`
Let, x = `sin theta => theta = sin^-1 x`
`therefore y = sin^-1 (2 sin theta sqrt(1 - sin^2 theta))`
`= sin^-1 (2 sin theta, cos theta)`
`= sin^-1 (sin 2 theta) ...(because sin 2 theta = 2 sin theta . cos theta)`
`y = 2 theta`
`=> y = 2 sin^-1 x`
`therefore dy/dx = 2 d/dx sin^-1 x`
`⇒ dy/dx= 2/sqrt(1 - x^2)`
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