Let f(x) be the function given by `f(x)=arc cos ((1-x^(2))/(1+x^(2))).Then`

Options

A. f(x) is everywhere differential such that `f'(x)=(2)/(1+x^(2))`
B. `f'(x)={{:(,(2)/(1+x^(2)),x gt0),(,(-2)/(1+x^(2)),x lt 0):}`
C. `f'(x)={{:(,(-2)/(1+x^(2)),x gt0),(,(2)/(1+x^(2)),x lt 0):}`
D. f'(x) exists at x=0

Answer

B

Explanation

We have
`f(x)=arc cos ((1-x^(2))/(1+x^(2)))`
`Rightarrow f(x)={{:(,2tan^(-1)x,"if "0 le x ltoo),(,-2tan^(-1)x,"if "-oo lt x le0):}`
Clearly, f(x) is everywhere continuous but not differentiable at x=0 such that `f(x)={{:(,(2)/(1+x^(2)),"if "0 lt x lt oo),(,(-2)/(1+x^(2)),"if -oo lt x lt 0):}`