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

[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))
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!

 

[mathman]

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

 

[Mark44]

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)

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 x^1/3?

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.