यदि `y=sin^(-1)((sqrt(1+x)+sqrt(1-x))/(2))` है, तो `(dy)/(dx)` बराबर है

Options

A. `(1)/(sqrt(1-x^(2)))`
B. `(x)/(sqrt(1-x^(2)))`
C. `(-1)/(2sqrt(1-x^(2)))`
D. इनमे से कोई नहीं

Answer

C

Explanation

दिया हैं

`y=sin^(-1)((sqrt(1+x)+sqrt(1-x))/(2))`

माना, `x=costheta`

तब,

`y=sin^(-1)((sqrt(1+costheta)+sqrt(1-costheta))/(2))`

`=sin^(-1)((sqrt(2)costheta2+sqrt(2)sintheta//2)/(2))`

`=sin^(-1)((1)/(sqrt(2))*"cos"(thetat)/(2)+(1)/(sqrt(2))*"sin"(theta)/(2))`

`=sin^(-1)[sin((pi)/(4)+(theta)/(2))]`

`y=(pi)/(4)+(theta)/(2)`

`rArry=(pi)/(4)+(1)/(2)cos^(-1)x`

`:." "(dy)/(dx)`

`=(d)/(dx)((pi)/(4))+(1)/(2)(d)/(dx)(cos^(-1)x)`

`=0+(1)/(2)((-1)/(sqrt(1-x^(2))))`

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