What is the form of the spectral derivative in the all-positive-frequency notation in DFT?
There is no requirement, it is a matter of choice.

Some conceptual clarity is required.  A sequence of sampled points does not have a derivative.  Derivatives come from continuous functions.

If you treat the inverse DFT definition as a continuous function, the result is a continuous interpolation function that will pass through all your sampled points.  Choosing different ranges (could be 6N to 7N-1 if you wanted) yields different interpolation functions each being differentiable.

This thread should give you some insight:

How to get Fourier coefficients to draw any shape using DFT?

The "simplest" or "most natural" range is using the negative frequencies, and if you have an even number of bins, the Nyquist bin should be split in half, one positive and one negative.