Question

`d/(dx)[sin^-1(xsqrt(1 - x) - sqrt(x)sqrt(1 - x^2))]` is equal to

Options

• `1/(2sqrt(x(1 - x))) - 1/sqrt(1 - x^2)`

• `1/sqrt(1 - {xsqrt(1 - x) - sqrt(x(1 - x^2))}^2`

• `1/sqrt(1 - x^2) - 1/(2sqrt(x(1 - x))`

• `1/sqrt(x(1 - x)(1 - x)^2`

Answer 1

`1/sqrt(1 - x^2) - 1/(2sqrt(x(1 - x))`

Explanation:

Let y = `d/(dx)[sin^-1(xsqrt(1 - x) - sqrt(x) sqrt(1 - x^2))]`

Put x = sin α and `sqrt(x)` = sin β

∴ `y = d/(dx)[sin^-1(sinalpha sqrt(1 - sin^2 beta) - sin beta sqrt(1 - sin^2 alpha))]`

= `d/(dx) [sin^-1 (sin alpha - beta)] = d/(dx) (alpha - beta)`

= `d/(dx) [sin^-1x - sin^-1 sqrt(x)]`

= `1/sqrt(1 - x^2) - 1/(2sqrt(x)sqrt(1 - x))`

= `1/sqrt(1 - x^2) - 1/(2sqrt(x(1 - x))`