# Analyzing Abs'

Version 12.1 on a Mac

The question about the derivative of Abs' has been asked several times, but all previous posts dealt with Abs'[x] and my question is about Abs'.

It seems to me that there are no complex-numbers issues with Abs', yet

 Abs'//N


does not simplify to 1.0. Is there a way to overcome this?

Edit:

For me, Abs appeared as a result of calling Norm on a real vector. How do I make it use RealAbs?

• 1 is a complex number, too. The problem is that Abs[z] is a complex function, not whether z is in the subset of real numbers. In terms of complex function theory, Abs'[z] is undefined. You can deal with it as it is dealt with elsewhere, such as ComplexExpand[Abs']. – Michael E2 Apr 6 at 18:56

One must somehow assert the domain $$\mathbb{R}$$ because the default domain is $$\mathbb{C}$$.

The symbol RealAbs is one such way:

RealAbs'

1


Also note that just because there are no explicit complex numbers doesn't mean the derivative exists over $$\mathbb{C}$$. Here's the derivative at z == 1 over both domains:

dq = DifferenceQuotient[Abs[z], {z, h}] /. z -> 1;

Limit[dq, h -> 0, Direction -> Complexes]

Indeterminate

Limit[dq, h -> 0, Direction -> Reals]

1

• Thanks, this clarifies the behavior. But how does one make the following work: NIntegrate[D[Norm[{Sin[t], Cos[t]}], t], {t, 0, 2 [Pi]}] – Wynne Apr 6 at 20:48
• You could define RealNorm[args__] := Norm[args] /. Abs -> RealAbs. – Chip Hurst Apr 6 at 21:17
• You could even remove some instance of Abs with RealNorm[args__] := Norm[args] /. {Abs[f_]^n_Integer?EvenQ :> f^n, Abs[f_] :> RealAbs[f]} – Chip Hurst Apr 6 at 23:37