If `y=cos^(-1)((2x)/(1+x^2)),t h e n(dy)/(dx)i s` `(-2)/(1+x^2)` for all `x` (b) `(-2)/(1+x^2)` for all `|x|<1` `2/(1+x^2)` for `|x|>1` (d) none of these

Options

A. `(-2)/(1+x^(2))` for all x
B. `(-2)/(1+x^(2))" for all "|x|lt 1`
C. `(2)/(1+x^(2))" for "|x|gt1`
D. none of these

Explanation

`y=cos^(-1)((2x)/(1+x^(2)))`
`"or "(dy)/(dx)=(1)/(sqrt(1-(4x^(2))/((1+x^(2))^(2))))(d)/(dx)((2x)/(1+x^(2)))`
`=-(1+x^(2))/(sqrt((1+x^(2))^(2)))(2(1+x^(2))-4x^(2))/((1+x^(2))^(2))`
`=-2(1+x^(2))/(|1-x^(2)|)(1-x^(2))/((1+x^(2))^(2))`
`=-2((1-x^(2))/(|1-x^(2)|))((1)/(1+x^(2)))`
`={{:((2)/(1+x^(2))",",if |x|gt1),((-2)/(1+x^(2))",",if|x|lt1):}`