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How to calculate the derivative in (0, ∞)?

  1. Oct 5, 2014 #1
    The function f: R → R is: f(x) =

    (tan x) / (1 + ³√x) ; for x ≥ 0,

    sin x ; for (-π/2) ≤ x < 0,

    x + (π/2) ; for x < -π/2
    _

    For the interval (0,∞), we are interested in f such that
    f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0
    f(x) = tan x / (1 + x¹ʹ³)

               (1 + x¹ʹ³)•sec²x − tan x  •  (⅓ x ⁻²ʹ³)
    f'(x) = ———————————————            ← by the quotient rule
                                (1 + x¹ʹ³)²

               (1 + ∛x) sec²x   −   (tan x  /  (3 ∛(x²))
    f'(x) = ———————————————                  ← ANSWER
                                (1 + ∛x)²

                                  OR

               3x²ʹ³ (1 + x¹ʹ³)•sec²x − tan x
    f'(x) = —————————————                  ← ANSWER
                         3x²ʹ³ (1 + ∛x)²

    >> From here I can't go any further...
    Any help is more than welcome!
     
    Last edited: Oct 5, 2014
  2. jcsd
  3. Oct 5, 2014 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    What are you trying to do? The answer is clumsy, but it looks correct.
     
  4. Oct 5, 2014 #3

    Mark44

    Staff: Mentor

    In the last line above, what you wrote doesn't make sense. You have x¹ʹ³. I can't tell what the symbol is between 1 and 3. Is that supposed to be x1/3?

    BTW, homework- or coursework-type problems should be posted in the Homework & Coursework sections, not in the technical math sections. I have moved this thread.
     
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