But if I copyright the prompt to the input of the AI, does that not also become a derivative work?

I do believe in the US (I could be wrong) that copyright does exist without registering it with the US government.

Wouldn't AI generated art be derivative work done by Google (or whoever) when creating their Gemini models? So then Google owns all gemini created ai artwork?

1. Model weights may not be copyrightable.

2. Copyright protects copying. Expressive elements from the original creative work (source code) exist in the byte code, thus it remains under the original copyright.

3. For a derivative work to be considered a newly copyrightable work (as opposed to a copy subject to the original's copyright), it must contain new substantive human creative expression (whether the original creator also has a copyright claim as well depends on degree of transformation).

Typically you need the permission of a copyright owner to create derivatives of their work.