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If f(x)=cot⁻¹ ((x^x-x^(-x))/(2)), then the value of f'(1) is

1. -1
2. log 2
3. 1
4. -log 2

Correct answer: -1

Solution

Let u(x)=(x^x - x^-x)/2, with u(1)=0 and u'(1)=( (x^x(1+ln x)) + (x^-x(1+ln x)) )/2 at x=1 = (1+1)/2 = 1. Then f'(x) = -u'/(1+u^2), so f'(1) = -1/(1+0) = -1.

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