When I put X^(1/3) into Y=, and go to nDeriv, and

Hello,
The derivative of the cubic root of x is, as you wll know, equal to (1/3)*x^(-2/3). Its limit as x approaches 0 is undefined. However the calculator uses approximate representations of numbers to calculate. And you can never be sure what it is going to give as results near singularities (poles of functions).

Calculators with Computer Algebra Systems can do better. To show you this, I am enclosing a screen capture showing you the correct result.

On this screen I defined a function u =f(v) where u is the derivative of cubic root of v. I stored 0 in the dummy variable v, and I evaluated the function u at v=0. You can see that the result displayed is undefined.
That this result is deemed correct seems to me just a matter of convention. We do not know, we cannot know, or we dont care to know? It will not make any of us lose sleep.

Hope it helps.