By using a difference function, as it's limit goes to zero, ... so w/ infintesimals, differentials, you are really making vectors, and so the differential of I wrt V will be a different type of quantity, than just dividing the instantanous values of I/V

 

Perhaps it’s just a gross simplification meant for napkin math.

 

If you post a copy of your circuit, I will show you how you derive the various voltages and currents.
Your second equation should be shown with a lower case "v" and "i" to indicate that it refers to small signal analysis.