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Question
Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`
Solution 1
`y = sin^-1(sqrt(1-x^2))`
Put x = cosθ
∴ θ = cos-1x
∴ `y = sin^-1(sqrt(1-cos^2theta))`
`=sin^-1(sqrt(sin^2theta))`
`=sin^-1(sin^2theta)`
= θ
= cos-1x
`therefore (dy)/(dx)=-1/(sqrt(1-x^2))`
Solution 2
`y = sin^-1(sqrt(1-x^2))`
Differentiating w.r.t. x
`(dy)/(dx)1/(sqrt(1-(sqrt(1-x^2))^2))(dy)/(dx)(sqrt(1-x^2))`
`=1/sqrtx^2 1/(2sqrt(1-x^2)) (-2x)`
`=(-1)/(xsqrt(1-x^2)) (x)`
`=(-1)/(sqrt(1-x^2))`
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