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

Options

A. `(sqrt(pi))/(6)`
B. `- sqrt((pi)/(6))`
C. `(1)/(sqrt(6))`
D. `(pi)/(sqrt(6))`

Answer

B

Explanation

We have , `f(x) = sqrt(1 + cos^(2) (x^(2)))` ...(i)
On differentiating both sides of Eq .(i) w.r.t.x, we get
`f'(x) =((d)/(dx)(1+ cos ^(1) (x)^(2)))/(2sqrt( 1 + cos^(2) x^(2)))=(- 2 sin x^(2) cosx^(2))/(sqrt(1 + cos^(2) x^(2)))(x)`
` rArr r'(x) = (- sin2 x^(2))/(sqrt(1 + cos^(2) x^(2)))(x)`...(ii)
On putting ` x = (sqrt(pi))/(2)` in Eq . (ii) , we get
`f'(sqrt(pi)/(2)) = - sqrt(pi)/(2) .(sin2((pi)/(4)))/(sqrt( 1 + cos^(2) (pi)/(4))) =- (sqrt(pi))/(2).(sin""(pi)/(2))/(sqrt((3)/(2)))=0sqrt((pi)/(6)`