Question

Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`

Answer 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))`

Answer 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))`