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SandBoxAbs

Edit detail for SandBoxAbs revision 3 of 3

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Editor: Bill Page Time: 2015/01/30 18:33:28 GMT+0
Note: 2nd derivative

changed:
-abs is often defined in terms of the positive square root
The *abs* function is often defined in terms of the positive square root

added:

The derivative of *abs* is often represented by the *signum* function
(or *sign*) and the derivative of *signum* is the Dirac *delta* function
which of course is not really a functions at all ...
So what is the 2nd derivative of *abs*?

\begin{axiom}
differentiate(abs(x),[x,x])
differentiate(Abs(x),[x,x])
\end{axiom}

This is how FriCAS currently differentiates the absolute value function.

fricas

(1) -> differentiate(abs(x),x)

$\label{eq1}\frac{abs \left({x}\right)}{x}$

(1)

Type: Expression(Integer)

The abs function is often defined in terms of the positive square root

fricas

Abs(x)==sqrt(x^2)

Type: Void

so comparison with the following is perhaps a surprize.

fricas

differentiate(Abs(x),x)

fricas

Compiling function Abs with type Variable(x) -> Expression(Integer)

$\label{eq2}\frac{x}{\sqrt{{x}^{2}}}$

(2)

Type: Expression(Integer)

Are these expressions really equivalent?

The derivative of abs is often represented by the signum function (or sign) and the derivative of signum is the Dirac delta function which of course is not really a functions at all ... So what is the 2nd derivative of abs?

fricas

differentiate(abs(x),[x,x])

$\label{eq3}0$

(3)

Type: Expression(Integer)

fricas

differentiate(Abs(x),[x,x])

$\label{eq4}0$

(4)

Type: Expression(Integer)