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Difficulty understanding methodology

Posted 5 months ago

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).

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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
POSTED BY: Gianluca Gorni

Thank you - did not know about Surd - exactly what I needed. Again, thank you for your help.


Posted 5 months ago

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 =)

POSTED BY: Denis Ivanov
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