This is how FriCAS currently differentiates the absolute value function.

differentiate(abs(x),x)

(1)

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

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

so comparison with the following is perhaps a surprize.

differentiate(Abs(x),x)

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

(2)

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?

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

(3)

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

(4)