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If f(x)=cot^(-1)((x^(x)-x^(-x))/(2)), th...

If `f(x)=cot^(-1)((x^(x)-x^(-x))/(2)),` then `f'(1)` equals

A

-1

B

1

C

`log_(e)2`

D

`-log_(e)2`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`f(x)=cot^(-1)((x^(x)-x^(-x))/(2))`
`implies" "f'(x)=-(1)/(1+((x^(x)-x^(-x))/(2))^(2)).(d)/(dx)((x^(x)-x^(-x))/(2))`
`implies" "f'(x)=(-2)/(4+(x^(x)-x^(-x))^(2)).(d)/(dx)(x^(x)-x^(-x))`
`implies" "f'(x)=(-2)/(4+(x^(x)-x^(-x))^(2)).(d)/(dx)(e^(xlogx)-e^(-xlogx))`
`implies" "f'(x)=(-2)/((x^(x)+x^(x))^(2))xx{e^(xlogx).(d)/(dx)(xlogx)-e^(-xlogx)(d)/(dx)(-xlogx)}`
`implies" "f'(x)=(-2)/((x^(x)+x^(-x))^(2)){x^(x)(1+logx)+x^(-x)(1+logx)}`
`implies" "f'(x)=(-2(1+logx))/((x^(x)+x^(-x))^(2)).(x^(x)+x^(-x))=(-2(1+logx))/(x+x^(-x))`
`implies" "f'(1)=(-2)/((1+1))=-1`
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