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Dr. Wolfgang Hintze
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Edit 15.09.14

There has been quite a lot of discussion here but no answer. I gather that the answers to similar topics referenced in the comments here are considered sufficient. These are:

Derivative of real functions including Re and Im

Symbolic derivatives are being calculated numerically

Because these references are pretty comprehensive I don't know if my question has contributed anything new, and the surprise was only on my side.

Let me nevertheless add some further observations which show that in some cases the documentation points out Possible Issues. But this is not done consistently. In one case WolframAlpha gives the expected result which MMA has refused to give.

1a) Abs'[0.1] is not evaluated.

But WolframAlpha "knows better":

WolframAlpha["Abs'[0.1]"];

(* -> 1 *)

1b) Trying to tell Mathematica that x is not complex but real (in which case the derivative is well defined)

Assuming[x \[Element] Reals, D[Abs[x], x]]

(* -> Derivative[1][Abs][x] *)

doesn't work either.

1c) Abs Possible Issues says: No series can be formed from Abs for complex arguments:

Series[Abs[x], {x, 0, 2}]

(* -> Abs[x] *)

For real arguments, a series can be found:

Series[Abs[x], {x, 0, 2}, Assumptions -> Element[x, Reals]]

$\begin{array}{ll} \{ & \begin{array}{ll} -x+O[x]^3 & x\leq 0 \\ x+O[x]^3 & \text{True} \\ \end{array} \\ \end{array}$

1d) UnitStep Possible Issues says: Differentiating Abs does not yield UnitStep:

D[Abs[t], t]

(* Derivative[1][Abs][t] *)
  1. HeavisideTheta / DiracDelta Possible Issues say:

The functions UnitStep and HeavisideTheta are not mathematically equivalent:

{HeavisideTheta[x], UnitStep[x]}
Integrate[D[%, x], x]

(* -> {HeavisideTheta[x], UnitStep[x]} *)

Only HeavisideTheta gives DiracDelta after Differentiation.

{HeavisideTheta[x], UnitStep[x], (Sqrt[x^2]/x + 1)/2, (Abs[x]/x + 1)/2};

D[%, x] // Together

$\left\{\text{DiracDelta}[x], \begin{array}{ll} \{ & \begin{array}{ll} \text{Indeterminate} & x==0 \\ 0 & \text{True} \\ \end{array} \\ \end{array} ,0,\frac{-\text{Abs}[x]+x \text{Abs}'[x]}{2 x^2}\right\}$

  1. Conjugate Possible Issues says: Differentiating Conjugate is not possible:

    D[Conjugate[t], t] (* Derivative[1][Conjugate][t] *)

A similar remark should be placed in the documentation under Possible Issues consistently.

  1. Sign' behaves similar as Abs'

    Sign'[0.1]

    (* Derivative[1][Sign][0.1] *)

    WolframAlpha["Sign[0.1]"]; (* -> 1 *)

Regards, Wolfgang

Regards, Wolfgang

Edit 15.09.14

There has been quite a lot of discussion here but no answer. I gather that the answers to similar topics referenced in the comments here are considered sufficient. These are:

Derivative of real functions including Re and Im

Symbolic derivatives are being calculated numerically

Because these references are pretty comprehensive I don't know if my question has contributed anything new, and the surprise was only on my side.

Let me nevertheless add some further observations which show that in some cases the documentation points out Possible Issues. But this is not done consistently. In one case WolframAlpha gives the expected result which MMA has refused to give.

1a) Abs'[0.1] is not evaluated.

But WolframAlpha "knows better":

WolframAlpha["Abs'[0.1]"];

(* -> 1 *)

1b) Trying to tell Mathematica that x is not complex but real (in which case the derivative is well defined)

Assuming[x \[Element] Reals, D[Abs[x], x]]

(* -> Derivative[1][Abs][x] *)

doesn't work either.

1c) Abs Possible Issues says: No series can be formed from Abs for complex arguments:

Series[Abs[x], {x, 0, 2}]

(* -> Abs[x] *)

For real arguments, a series can be found:

Series[Abs[x], {x, 0, 2}, Assumptions -> Element[x, Reals]]

$\begin{array}{ll} \{ & \begin{array}{ll} -x+O[x]^3 & x\leq 0 \\ x+O[x]^3 & \text{True} \\ \end{array} \\ \end{array}$

1d) UnitStep Possible Issues says: Differentiating Abs does not yield UnitStep:

D[Abs[t], t]

(* Derivative[1][Abs][t] *)
  1. HeavisideTheta / DiracDelta Possible Issues say:

The functions UnitStep and HeavisideTheta are not mathematically equivalent:

{HeavisideTheta[x], UnitStep[x]}
Integrate[D[%, x], x]

(* -> {HeavisideTheta[x], UnitStep[x]} *)

Only HeavisideTheta gives DiracDelta after Differentiation.

{HeavisideTheta[x], UnitStep[x], (Sqrt[x^2]/x + 1)/2, (Abs[x]/x + 1)/2};

D[%, x] // Together

$\left\{\text{DiracDelta}[x], \begin{array}{ll} \{ & \begin{array}{ll} \text{Indeterminate} & x==0 \\ 0 & \text{True} \\ \end{array} \\ \end{array} ,0,\frac{-\text{Abs}[x]+x \text{Abs}'[x]}{2 x^2}\right\}$

  1. Conjugate Possible Issues says: Differentiating Conjugate is not possible:

    D[Conjugate[t], t] (* Derivative[1][Conjugate][t] *)

A similar remark should be placed in the documentation under Possible Issues consistently.

  1. Sign' behaves similar as Abs'

    Sign'[0.1]

    (* Derivative[1][Sign][0.1] *)

    WolframAlpha["Sign[0.1]"]; (* -> 1 *)

Regards, Wolfgang

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Dr. Wolfgang Hintze
Dr. Wolfgang Hintze
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Strange behaviour of MMA in derivatives of some standard functions

There are some peculiar things to be discovered in derivatives of some standard functions in MMA:

Strange behaviour

Example 1: Abs

We have

Abs[0.1]

(* Out[72]= 0.1 *)

But

Abs'[0.1]

(* Out[73]= Derivative[1][Abs][0.1] *)

Same thing if we avoid writing the prime:

D[Abs[x], x] /. x -> 0.1

(* Out[110]= Derivative[1][Abs][0.1] *)

That is, the derivative of Abs[] has no numerical value. When plotted, there's nothing to be seen.

The derivative should of course be, or at least behave like, Sign[x] .

Example 2: Sign

Sign[0.1]

(* Out[78]= 1 *)

Sign'[0.1]

(* Out[79]= Derivative[1][Sign][0.1] *)

That is, the derivative of Sign[] has no numerical value. When plotted, there's nothing to be seen.

Example 3: Floor

Floor[0.1]

(* Out[80]= 0 *)

But which value has the derivative at 0 ?

First possibility:

Floor'[0]

(* Out[89]= Derivative[1][Floor][0] *)

No value.

Second possibility:

Floor[0.]

(* Out[90]= 0 *)

Hence the numerical value seems to be 0.

No, bad luck, wrong guess! Look at that

Third possibility:

N[Floor'[0]]

(* Out[93]= 36.2120995105236639865598801739718338716778459859 *)

Furthermore:

Table[{x, N[Floor'[x], 5]}, {x, 0, 1/2, 0.05}]

(* Out[96]= {{0., 36.212}, {0.05, 12.5946}, {0.1, -4.29432}, {0.15, 
  1.61679}, {0.2, -0.532708}, {0.25, 0.13901}, {0.3, -0.0263362}, {0.35, 
  0.00318987}, {0.4, -0.000184538}, {0.45, 0.}, {0.5, 0.}} *)

Strange ocillatory behaviour for a quantity which should be = 0 throughout. It seems to be defined via a Fourier series.

Remedy ?: do it yourself !

Finally, let's create the derivative by ourselves as it is originally defined:

floorPrime[x_] := Limit[(1/h) (Floor[x + h] - Floor [x]), h -> 0]

Plot[0.1 + floorPrime[x], {x, -1, 1}, PlotRange -> {0, 0.2}]

(* Picture snipped *)


absPrime[x_] := Limit[(1/h) (Abs[x + h] - Abs[x]), h -> 0]

Plot[absPrime[x], {x, -1, 1}, PlotRange -> {-2, 2}]

(* Picture snipped *)

Ok, everything fine.

But why has MMA such problems with its own standard operation ' (or D[]) in this class of functions? Please explain.

Regards, Wolfgang