Why does the function y = | x | have no derivative in x = 0?

Why does the function y = | x | have no derivative in x = 0?

Because it's not smooth at x = 0
We can also consider that when x > 0, the derivative of y = | x | = x at x = 0 = 1,
When x

Does the non differentiability of a function mean that the derivative of a function does not exist

For example, y = | x | is not differentiable at (0.0) point, because its left and right limits are not the same, so there is no tangent at that point

When does a function have no derivative?

The most fundamental method is based on the definition of derivative
Intuitively, it is the point where the tangent does not exist
Here is a specific example:
The derivative of F (x) = | x |, x = 0 does not exist