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Differential coefficient of sec^(-1) (1...

Differential coefficient of ` sec^(-1) (1)/(2x^(2) -1)` with respect to
`sqrt(1 -x^(2) ) " at x " = (1)/(2) ` is equal to

A

2

B

4

C

6

D

1

Text Solution

Verified by Experts

The correct Answer is:
B
Let ` u = sec^(-1) (1)/(2x^(2) -1) "and " v = sqrt(1 - x^(2)) `. Then
`(du)/(dx) = (1)/(((1)/(2x^(2) -1)))(1)/(sqrt(((1)/(2x^(2) -1))^(2)-1))xx((1)/(2x^(2) -1)) xx 4 `
` rArr (du)/(dx) = - (4x)/(sqrt( 1-(4x^(4) +1 - 4x^(2))))`
` rArr (du)/(dx) = (-4x)/(sqrt( 4x^(4) - 4x^(2)))=(-2)/(sqrt(1-x^(2)))`
and `(du)/(dx) = (-2x)/(sqrt( 1-x^(2)))=(-x)/(sqrt( 1 -x^(2)))`
Thus , `(du)/(dx) = (-2x)/(sqrt( 1-x^(2))) and (dv)/(dx)=(-x)/(2sqrt( 1 -x^(2)))`
`rArr (du)/(dx) = (2)/(x) rArr ((du)/(dv_(2)))_(x=1//2) = 4 `
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