I am trying to understand the transconductance of the mosfets. By definition it is the change in the drain current divided by the gate-source voltage. However, some sources shows it with a derivative equation such as:

But sometimes it is shown without the derivative operation.

Some of the forums I found says that derivative is not used in the case of a small signal analysis. But it doesn't make sense to me and also Razavi's Design of Analog CMOS Integrated Design shows this equation with derivative in the case of a small signal analysis. So, is there anyone to clarify this? Thanks for your help.