Find the derivative of `sec^(-1)((1)/(2x^(2)-1))" w.r.t. "sqrt(1-x^(2))" at "x-(1)/(2).`

Options

A. -4
B. 4
C. -1
D. none of these

Answer

A

Explanation

Let `y=cosec^(-1)((1)/(2x^(2)-1))` and `z=sqrt(1-x^(2))`
We have,
`y=cosec^(-1)((1)/(2x^(2)-1))`
`y=sin^(-1)(2x^(2)-1)=(pi)/(2)-cos^(-1)(2x^(2)-1)`
`y={{:((pi)/(2)-2cos^(-1)x" ,"," if "0lexle1),(-(3pi)/(2)+2cos^(-1)x","," if "-1lexle0.):}`
`(dy)/(dx)={{:((2)/(sqrt(1-x^(2)))","," if "0ltxlt1),((2)/(sqrt(1-x^(2)))","," if "-1ltxlt0):}`
and, `x=sqrt(1-x^(2))implies(dz)/(dx)=(-x)/(sqrt(1-x^(2)))" for all "x in(-1,1)`
`(dy)/(dz)=(dy//dx)/(dz//dx)={{:(-(2)/(x)","," if "0ltxlt1),(-(2)/(x)","," if "-1ltxlt0):}`

`implies" "((dy)/(dz))_(x=1//2)=-4`