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

Options

A. `(1)/(2sqrt(x(1-x)))-(1)/(sqrt(1-x^2))`
B. `(1)/(sqrt(1-{xsqrt(1-x)-sqrt(x(1-x^2))}^2))`
C. `(1)/(sqrt(1-x^2))-(1)/(2sqrt(x(1-x)))`
D. `(1)/(sqrt(x(1-x)(1-x)^2))`

Answer

C

Explanation

Let `y=sin^(-1)[xsqrt(1-x)-sqrt(1-x^2)]`
Put `x=sin alpha and sqrt(x)= sin beta`
`therefore y=sin^(-1)[sinalphasqrt(1-sin^(2)beta)-sinbetasqrt(1-sin^(2)alpha)]=sin^(-1)[sin(alpha-beta)]=alpha-beta`
`=sin^(-1)x-sin^(-1)sqrt(x)`
`rArr (dy)/(dx)=(1)/(sqrt(1-x^2))-(1)/(sqrt(1-x)).(1)/(2sqrt(x))=(1)/(sqrt(1-x^2))-(1)/(2sqrt(x(1-x)))`