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I generally think of the
derivative of something with respect to time as the rate of change of
that something.
In the case of a simple circuit such as a resistor
connected across a battery, the current is typically written as
I=dq/dt.
The question is, what is the q that is changing?
In the
case of a different circuit, where charge is piling up on the plate
of a capacitor, it's just the charge on that capacitor plate; no
problem there.
The best I can come up with is that q represents the total amount of
charge that has crossed a boundary in one of the two possible
directions, where the boundary is the point in the circuit to which
the current pertains,
and for charge that has made it all the way
around the loop, double counting is allowed; each time that charge
passes through boundary it contributes to the q.
But in the case of a resistor connected across a
battery, nothing has the charge q that is changing with time. We
write the expression as if q were a state variable, but nothing has
the corresponding state.