Question

If f(x) cot^–1 |[(x^x – x^x)/2]| the f'(1) =_________

(a) – 1 

(b) 1 

(c) log_e2 

(d) – log_e2

Answer 1

The correct option (a) – 1   

Explanation:

 f(x) = cot^–1[(x^x – x^–x)/2]

Let u = x^x hence log u = x log x

∴ (1/u) ∙ (du/dx) = (1/x) ∙ x + log x ⇒ (du/dx) = x^x (1 + log x)

Let V = x^–x hence (dv/dx) = – x^–x (1 + log x)

∴ f'(x) = [(– 1)/(1 + {(x^x – x^–x)/2}²)] ∙ (d/dx) [(x^x – x^–x)/2]

f'(x) = [(– 4)/(x^2x + x^–2x + 2)] [(1/2) [x^x + x^–x] [1 + log x]]

∴ at = x = 1, we get

f'(1)  = [(– 4)/(1 + 1 + 2)] [(1/2) (2) (1 + log1)]

= – 1 [1]

= – 1