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If f(x)=cos^(-1)[(1-(logx)^2)/(1+(logx)^...

If `f(x)=cos^(-1)[(1-(logx)^2)/(1+(logx)^2)]` , then the value of `f'(e)` is equal to........

A

`(1)/(e)`

B

`(2)/(e^(2))`

C

`(2)/(e)`

D

`1`

Text Solution

Verified by Experts

The correct Answer is:
A
We have
`f(x)=cos^(-1)[(1-(logx)^(2))/(1+(logx)^(2))]`
`f(x)=2tan^(-1)(logx)[becausecos^(-1)((1+x^(2))/(1+x^(2)))=2tan^(-1)x]`
`f'(x)=(2)/(1+(logx)^(2))xx(1)/(x)`
`f'(e)=(2)/(1+1)xx(1)/(e)=(1)/(e)`
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