Differentiate `sin^(-1)((2x)/(1+x^2))` with respect to `tan^(-1)((2x)/(1-x^2)),` if `-1

Options

A. 1 for all x
B. `1" for "|x|gt1" and"-1" for "|x|lt1`
C. `1" for "|x|lt1" and"-1" for "|x|gt1`
D. `1" for "|x|le1" and"-1" for "|x|gt1`

Answer

C

Explanation

Let `y=tan^(-1)((2x)/(1-x^(2)))" and "x=sin^(-1)((2x)/(1+x^(2))).`
Then,
`y={{:(2tan^(-1)x" ,","if"-1ltxlt1),(2tan^(-1)x-pi","," if "xgt1),(2tan^(-1)x+pi","," if "xlt-1):}`
and,
`z={{:(2tan^(-1)x" ,","if"-1lexle1),(pi-2tan^(-1)x" ,"," if "xgt1),(-pi-2tan^(-1)x","," if "xlt-1):}`
`:." "(dy)/(dz)={{:(1" ,"," if"-1ltxlt1),(-1","," if "|x|gt1):}`