If one takes the derivative of the function (x-4)^(1/3) you get 1/(3(x-4)^(2/3)). If you evaluate this derivative at x=-4, Mathematica’s answer is -1/(12(-1)^(1/3)). I don’t understand why the answer isn’t 1/12 since1/(3(-4-4)^(2/3) would be 1/(3(-8^2)^(1/3) which then would be 1/(3(64)^(1/3) resulting in 1/(3(4).
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:
- For real root, as mentioned Gianluca Gorni one should use
Surd[]. - For principal root we need
ComplexExpand[D[(x - 4)^(1/3), x] /. x -> -4] - But in fact, mentioned functions have 3 branches. Getting all their values is a separate interesting task =)
If you have in mind the cubic root in the real sense, then you should use Surd, not Power:
D[Surd[x - 4, 3], x] /. x -> -4
Thank you - did not know about Surd - exactly what I needed. Again, thank you for your help.
Dennis