calculus and analysis - Simplifying the derivative of $|x|$

Context

In[855]:= D[Abs[x], x] /. x -> 1

Out[855]= Derivative[1][Abs][1]

In[856]:= D[x, x] /. x -> 1

Out[856]= 1

Question

Why is Derivative[1][Abs][x] not simplifed to 1 when x -> 1?

Answer

$\mathrm{abs}(z)$ defined on the set of complex numbers $\mathbb{C}$ is not a holomorphic function because it violates the Cauchy-Riemann conditions, and the derivative is not well defined. $\mathrm{abs}(x)$ defined on the set of real numbers $\mathbb{R}$ is differentiable everywhere except at $x=0$.

Mathematica treats Abs[x] as a function defined on complex numbers and so will not simplify Abs'[x] when it occurs in expressions. However, you can force it to simplify by explicitly stating that the argument is real as follows:

FullSimplify[Abs'[x], x ∈ Reals]
(* Sign[x] *)

Also see this question for a very closely related problem.

Comments

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