Problem in dealing with Sign function

Question

Let my function be $f(x)=e^{|x|}$. I wish to find $f''(x)$, this can be done as-

$f'(x)=$sgn$(x)e^{|x|}$

$f''(x)=$sgn$(x)^2e^{|x|} =e^{|x|}$

In Mathematica, I am trying to plot $f''(x)$, it presents an empty plot.

f[x_] := E^Abs[x];
Plot[f[x], {x, -1, 1}]

g[x_] = D[f[x], {x, 2}]
Plot[g[x], {x, -1, 1}]

Could anyone please locate the error.

Carl Woll · Accepted Answer · 2018-10-10T05:50:58.360

3

You could use RealAbs instead of Abs:

FullSimplify[
    D[Exp[RealAbs[x]],x,x],
    x ∈ Reals
]

E^RealAbs[x]

Another possibility is to use ComplexExpand:

Simplify[
    D[ComplexExpand[Exp[Abs[x]]],x,x],
    x ∈ Reals
]

E^Abs[x]

edited Oct 10 '18 at 05:50

answered Oct 10 '18 at 05:47

Carl Woll

RealAbs[] is not working in my computer :( – Schrodinger Oct 10 '18 at 05:50
Alas! Again is not working :( – Schrodinger Oct 10 '18 at 05:53
@SachinKumar What version of Mathematica? – Carl Woll Oct 10 '18 at 05:54
It is version 9.0 – Schrodinger Oct 10 '18 at 05:58
RealAbs was introduced in M11.1. The ComplexExpand version should work in M9. What result do you get? – Carl Woll Oct 10 '18 at 05:59
Yes, It gives, E^Abs[x], But if I try to plot Plot[D[ComplexExpand[Exp[Abs[x]], x, x], x [Element] Reals], {x, 0, 2}], then it gives an error. May be I am trying to plot in a wrong way. – Schrodinger Oct 10 '18 at 05:59
Of course that doesn't work. For example, see 1301 – Carl Woll Oct 10 '18 at 06:01
okok, I will see that. anyway, thanks for the help. – Schrodinger Oct 10 '18 at 06:04
1

@Sachin, Exp[Piecewise[{{x, x > 0}, {-x, x < 0}}, 0]] should also work in version 9 for your purposes. – J. M.'s missing motivation Oct 10 '18 at 10:53

score 0 · Answer 2 · answered Oct 10 '18 at 07:24

0

An easy workaround for real x is to use Sqrt[x^2] instead of Abs[x]

Plot[D[Exp[Sqrt[x^2]], x] // Evaluate, {x, -1, 1}]

answered Oct 10 '18 at 07:24

Ulrich Neumann

2 Answers2