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How to simplify Sign'

Severin Posta, CTU
Posted 4 years ago

How can I force Mathematica to simplify the derivative of Sign? The following

FullSimplify[D[Sign[x], x], x > 0]

does not work. All it produces is

Sign'[x]

(On the other hand,

FullSimplify[D[Abs[x], x], x > 0]

goes through.)
POSTED BY: Severin Posta
7 Replies
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Sign is a programming construct, not an analytic function. However, you can define a generalized function with similar properties and differentiate that:

sign[x_] := 2 HeavisideTheta[x] - 1
sign'[x]
(* 2 DiracDelta[x] *)
POSTED BY: John Doty

You could get around this by:

    sgn[x_] := Piecewise[{{-1, x < 0}, {0, x == 0}, {1, x > 0}}]

    sgn'[x]
(* Piecewise[{{0, x < 0 || x > 0}}, Indeterminate]  *)

    D[sgn[x], x]
(* iecewise[{{0, x < 0 || x > 0}}, Indeterminate] *)

I'm showing the InputForm of each output. In a notebook you'll actually see the usual nice display, with a big left brace enclosing the separate cases on each line.
POSTED BY: Murray Eisenberg

Or do it:

 D[PiecewiseExpand[Sign[x], Assumptions -> x \[Element] Reals], x]
 (*Piecewise[{{0, x < 0 || x > 0}}, Indeterminate]*)
POSTED BY: Mariusz Iwaniuk
Severin Posta
Severin Posta, CTU
Posted 4 years ago

Interesting. Thanks.
POSTED BY: Severin Posta 
PiecewiseExpand[Sign[x], Assumptions -> x > 0]

(* 1 *)

Regards M.I
POSTED BY: Mariusz Iwaniuk
Severin Posta
Severin Posta, CTU
Posted 4 years ago

Thanks. However I need to simplify Sign', not Sign.
POSTED BY: Severin Posta 
ComplexExpand[Re[D[Sign[x], x]], TargetFunctions -> Conjugate]
(* 0 *)
POSTED BY: Mariusz Iwaniuk
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