Both your function and it's derivative may have complex values for
$x\lt4$.
Thanks to Gianluca Gorni, I’ve worked out the details of complex roots in Mathematica.
In fact,
$(-1)^{1/3}$ has 3 values, or roots. All can be found with:
In:= ComplexExpand /@ (x /. Solve[x^3 == -1, x])
Out :=
$\left\{-1,\frac{1}{2}+\frac{i \sqrt{3}}{2},\frac{1}{2}-\frac{i \sqrt{3}}{2}\right\}$
So we have 3 cases:
1. For real root, as mentioned Gianluca Gorni one should use Surd[].
2. For principal root we need ComplexExpand[D[(x - 4)^(1/3), x] /. x -> -4]
3. But in fact, mentioned functions have 3 branches. Getting all their values is a separate interesting task =)
