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

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1. Oct 5, 2014

SixBooks

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))
(1 + ∛x)²

OR

3x²ʹ³ (1 + x¹ʹ³)•sec²x − tan x
3x²ʹ³ (1 + ∛x)²

>> From here I can't go any further...
Any help is more than welcome!

Last edited: Oct 5, 2014
2. Oct 5, 2014

mathman

What are you trying to do? The answer is clumsy, but it looks correct.

3. Oct 5, 2014

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.