The derivative of |x| is sgn x — plus one on the right, minus one on the left. At the origin
there is no derivative at all, and that gap is the interesting part.
Why the limit fails
Approaching from the right, the difference quotient is +1 at every step. From the left it is −1. The
one-sided limits disagree, so the two-sided limit does not exist. The graph has a corner rather than a
tangent, and a corner has no slope to report.
Why the calculator still answers
The engine returns sgn(u)·u', which is correct wherever u is not zero — that is, everywhere except a
set of isolated points. Refusing to answer at all would be less useful than answering with a stated
exception, so the rule page carries the caveat rather than the code carrying a special case.
The one that surprises people
x|x| is differentiable at the origin. Multiplying by x irons the corner out, and the derivative
2|x| touches zero smoothly.