dq/dt is the instantaneous flow of charge past a point. dt is the time interval over which the charge is counted and is made arbitrarily small. In fact dt approaches 0 in a mathematical sense. The reason for taking small slices (samples) of q is that the function is really i(t) = dq/dt. The current i is varying with time and we don't want to miss oddities by having too wide a sample.
This example may not help but consider charging a capacitor through a resistor from a battery. Beginning with no charge on the capacitor, once we close a switch and charge (current) begins to flow, the voltage on the capacitor will increase. Here's the point: The dq/dt value changes with the difference in voltage across the resistor - in other words, the difference between the battery voltage and the instantaneous voltage on the capacitor causes a current flow through the resistor and the resistor limits the current. If I use a small resistor, I get a high charge current.
This is an important concept and there is a lot to learn from the charge and discharge equations. If you make the time constant (R*C) long enough, say several seconds, you can actually watch the capacitor voltage on a DMM. To be fair, it is easier to see on an analog meter. You will see a large change in voltage early in the charge cycle when the capacitor isn't holding any charge and you will see a very small change when the capacitor voltage is nearly equal to the battery voltage. The small change in charge is exactly the same as saying the current is small since current is defined in terms of charge flowing past a point.
Attached is a graph of the charge and discharge of a capacitor scaled to 1V. You can multiply the values by any battery voltage you want. The time constant T (called tau) is 0.1 seconds. You will note that the capacitor charges to 63% of the battery voltage in the first T seconds. Funny thing, it charges to 63% of the difference between battery voltage and capacitor voltage in EVERY T interval. In the second interval, there is 37% voltage difference so we move .63 * 37 or 23%. Now, at the end of 2T seconds, we are at 63 + 23 or 86%.In the 3rd interval, we move up 63% of the remaining 14%. By 6 T intervals, we are essentially at 100% but, mathematically, we never get to 100%. We just get close enough for engineers.
The graph is based on 1000 ufd and 100 Ohms, Tau = R * C = 0.1 seconds. Here is a table of Tau versus %Percent Charge:
Tau = 0 Percent Charge = 0
Tau = 1 Percent Charge = 63
Tau = 2 Percent Charge = 86
Tau = 3 Percent Charge = 95
Tau = 4 Percent Charge = 98
Tau = 5 Percent Charge = 99
Tau = 6 Percent Charge = 100
The equations
Vchg = V0 * (1 - e(-t/Tau));
Vdis = V0 * (e(-t/Tau));
V0 is the battery voltage, Tau = R * C as above
If you want to try these on a calculator, just let -t/Tau be neat numbers like -1..-6
Yes, I know this is a bit off the wall but just about everything you need to know about charge on a capacitor and the rate of change of charge (dq/dt) is covered in this example. You can Google for 'capacitor charge' and get far better explanations. If this is too far afield right now, just copy it off and save it for another time.