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Differentiate sec^(-1)(1/(2x^2-1)), 0ltx...

Differentiate `sec^(-1)(1/(2x^2-1))`, `0ltxlt1/(sqrt(2))` with respect to `x`

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Verified by Experts

Given that,
`y=sec^(-1)(1/(2x^2-1))`
`0ltxlt1/(sqrt(2))`
`x=cos theta`
so, `y= sec^-1(1/(2cos^2theta-1))`
`=>y=sec^-1 (1/cos2theta)`
`=>y=sec^-1 (sec 2theta)`
`0ltxlt1/(sqrt(2))`
`=>0lt cos theta lt1/(sqrt(2))`
`=>pi/4 lt theta lt pi/2`
`=>pi/2 lt 2theta lt pi`
`y=2theta`
`=>y=2 cos^-1 x`
`(dy)/(dx)=-2/sqrt(1-x^2)`
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