Question

If `f(x)=sin^(-1) ((2x)/(1+x^2))` then `f(x)` is differentiable on
A. [-1,1]
B. R-{-1,1}
C. R-(-1,1)
D. None of these

Answer 1

Correct Answer - A
Given `f(x)=sin^(-1)((2x)/(1+x^(2)))`
On differentiating w.r.t.x. we get
`f(x)=(1)/(sqrt(1-((2x)/(1+x^(2)))^(2)))xx(d)/(dx)((2x)/(1+x^(2)))`
`=(1+x^(2))/(sqrt((1-x^(2))^(2)))xx(2(1-x^(2)))/((1+x^(2))^(2))`
`(2)/(1+x^(2))xx(1-xx^(2))/(|1-x^(2)|)`
`={{:(,(2)/(1+x^(2)),"if "|x|lt1),(,-(2)/(1+x^(2)),"if "|x|gt1):}`
`therefore` f(x) does not exist for `|x|=1, ie, x=pm1`
Hence f(x) is differentiable on `R-{(-1,1)`