For this question, let f(x)=|x|.
I found this answer saying that the derivative of the absolute value function is the signum function. In symbols,
ddx|x|=sgn(x).
I know that
f′(x)=x|x|
using the chain rule. Notice that this is well-defined for x≠0. However, the definition of the signum function is
sgnx={−1for x<00for x=01for x>0.
This will be my question:
Are x/|x| and sgnx the same derivative of |x|?
