You will have to put a bit of work into thinking about it, if you don't already see it. At the risk of being more rude than effective, I spelled it out pretty clearly above if you understand autodiff, which requires an understanding of the chain rule, which most people first learn about in a calculus class.

If you want to claim that autodiff is a replacement for all our differentiation needs, then you should try to understand when it doesn't work.

The differentiation of a fractional power (fraction less than 1.) involves a negative fractional power. 0 raised to some negative power involves dividing by zero. This gives you some kind of NAN in, e.g., vanilla languages such as Python. The "issue with automatic differentiation" is really an issue with dividing by zero. Graph x(1./3.). It's slope is vertical at zero.

I don't think this is right. An autodiff package will likely report +inf for the derivative of cuberoot at 0.

The problem is really that there are a lot of ways the function can have infinite slope. The sqrt function also has infinite slope at 0.

Dividing by zero isn't where the NaN comes from.

My mistake then. I did not intend to try and be so precise. I was thinking loosely. I have a very simple autodiff tool hand rolled (in Python) here to play with. I raise 0 to a negative exponent in the gradient of the cube root at 0. I do not get a NAN. Strictly speaking I get this:

"ZeroDivisionError: 0.0 cannot be raised to a negative power"

Is that satisfying? I do not claim anything more specific here. Sorry to be imprecise.