What is preventing Mathematica from simplifying $-\frac{1}{3}\frac{d}{dx}\ln|1+x^{-3}|$ even when assuming $x\in\mathbb{R}$?

Question

For the input

D[-1/3*Log[Abs[1 + x^(-3)]], x]

the output is

Derivative[1][Abs][1 + 1/x^3]/(x^4*Abs[1 + 1/x^3])

so I put

FullSimplify[D[-1/3*Log[Abs[1 + x^(-3)]], x], Element[x, Reals]]

to tell that Abs only takes real number arguments, but still the output is

Derivative[1][Abs][1 + 1/x^3]/(x^4*Abs[1 + 1/x^3])

What is preventing Mathematica from simplifying the output to

1/(x^4 + x)

and how can I get the expected result?

Try Simplify[D[-1/3*Log[RealAbs[1 + x^(-3)]], x], x \[Element] Reals]. The complex Abs[x] is not complex-differentiable at any x. (Alternatively, you can use ComplexExpand.) — Michael E2, Jun 25 '23 at 14:57
Does this answer your question? Fullsimplify does not simplify Abs'[1.-a] Also: Derivative of the Absolute Value Function in Mathematica looks very different from the actual answer — Domen, Jun 25 '23 at 14:58
Abs on the real line is the integral of the Sign distribution: Abs[x] =Integrate[Sign[x],{x,0,x]]. As a complex function, its differentiable exept at z=0 but not complex differentiable, it has the simple form Abs[z]= (z Conjugate[z])^(1/2). On the real line Sqrt(x^2). The ( #^2)^(1/2)& function is working as expected with the jump over the imaginary axis. — Roland F, Jun 26 '23 at 06:08

0 Answers0