Does the derivative for |x| not exist at 0?
@loki, you should consider the derivative from the left side and the derivative from the right side. consider those, then reconsider the idea of derivative of |x| at 0
$$f \left( x \right) = \left| x \right|, f \prime(x)$$
from the right, the function resembles y=x, from the left y=-x
Yes, the derivative of |x| does not exist at zero, look guy when we draw the graph of it you will have a sharp point at the origin differentiability guarantees the smooth points remember the surface of the jack fruit.
you can draw as many tangent line at x=0, that's why at x=0,|x| is not differentiable.
The derivative does not exist at sharp corners or cusps