Expression for the real 7/3 power

Question

I need an expression for the real 7/3 power of a real-valued function, i.e., a reformulation of

f[x_] := g[x]^(7/3)

that works for negative values of g[x]. This function and its first and second derivatives are used in heavy computation (i.e. in objectives to FindMinimum) and need to be well-defined at all values of x. In particular, I've tried the following, which don't work:

f[x_] := g[x]^(7/3)

which gives complex values at negative values of g[x], and

f[x_] := Surd[g[x],3]^7

which gives a "Infinite expression" error when its derivative is evaluated at 0.

Answer 1

To some extent, this is understandable given the formula for the derivative of the Surd.

D[Surd[x^4, 3], x] // InputForm
(* Out: (4*x^3)/(3*Surd[x^4, 3]^2) *)

Of course, this formula is good, except when $x=0$. Computer algebra systems frequently return this type of "generic" result. If you really need to work at zero, perhaps something like the following will work:

Clear[f, fp];
f[x_] = CubeRoot[x^7];
fp[x_] := f'[x] /; x != 0;
fp[0] = fp[0.0] = 0;

This should define a function fp representing your derivative that might work as expected. It also might depend on what you need to do with it.