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If f(x)=sqrt(1+cos^2(x^2)),t h e nf^(pri...

If `f(x)=sqrt(1+cos^2(x^2)),t h e nf^(prime)((sqrt(pi))/2)` is

A

`(sqrt(pi))/(6)`

B

`- sqrt((pi)/(6))`

C

`(1)/(sqrt(6))`

D

`(pi)/(sqrt(6))`

Text Solution

Verified by Experts

The correct Answer is:
B
Given , `f(x) = sqrt( 1 + cos^(2) (x^(2)))`...(i) On differentiating both sides of Eq. (i) w.r.t.x, we get
`f'(x) = (-2 sin x^(2) cos x^(2))/(2sqrt( 1 + cos^(2) (x^(2))))(2x)`
`rArr f'(x)= (-sin 2x^(2))/(sqrt(1 + cos^(2) (x^(2))))(x)`
` therefore f'(sqrt(pi)/(2))= - (sqrt(pi))/(2) . (sin 2 ((pi)/(4)))/(sqrt( 1 + (1)/(2)))= - sqrt((pi)/(6))`
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