a-maths~~!

(1)
f(x) = ln(tan x)
f'(x) = (sec x)^2 / tan x
x = π/6, f'(π/6) = [sec (π/6)]^2 / tan (π/6)
= (√3/2)^2 / (1/√3)
= 3√3 / 4
(2)
f(x) = arcsin (1/x)
sin (f(x)) = 1/x
f'(x) [cos (f(x))] = -1/x^2
f'(x) = -1 / [x^2 cos (f(x))]
= -1 / [x^2 (x / √(x^2 - 1)]
= - √(x^2 - 1) / x^3
when x = 2, f'(x) = - √(2^2 - 1) / (2^3) = -√3 / 8
(3)
f(x) = √(1 + (ln x)^2)
f'(x) = 0.5(2 ln x) (1/x) / √(1 + (ln x)^2)
= ln x / (x √( 1 + (ln x)^2)
when x = e,
f'(x) = ln e / (e √(1 + (ln e)^2) = 1 / (√2 e)


2008-07-20 08:07:31 補充：
Revised question 2

(2)

f(x) = arcsin (1/x)

sin (f(x)) = 1/x

f'(x) [cos (f(x))] = -1/x^2

f'(x) = -1 / [x^2 cos (f(x))]

= -1 / [x^2 ( √(x^2 - 1) / x]

= - 1 / [x√(x^2 - 1)]

when x = 2, f'(x) = - 1 / 2√(2^2 - 1) = -1 / 2√3 = -√3 / 6