Current as the time derivative of the charge I have been told that the current $i$ can be defined as
$ i = \displaystyle\frac{dq}{dt} $,
where $q$ is the charge and $t$ is the time. 
I do not understand this definition because, if the charges are moving so that the net charge remains constant in an infinitesimally thin cross-section of a wire, $q$ is constant with time and hence $dq/dt = 0$. That result would mean that no constant current can exist unless the charge change has a linear dependence with time (i.e. $q = q(t) \propto t$).
As I assume my reasoning is wrong, where is my mistake?
Thank you.
 A: Electric current can be understood through the analogy of water flow.
Just as the 'water current' in a river is the amount of water which passes a point in a given time interval, similarly, the electric current can be understood as the amount of electric charge passing through an area in a given time interval.
Of course, the analogy is not entirely correct, since charges can be either positive or negative, but the analogy is useful if we 'define' the electric current in terms of the 'flow' of positive charges

So, for example, considering the electric current through a wire of cross section $A$, the amount of charge ${\Delta q}$ passing through this surface in a time interval ${\Delta t}$ allows us to 'define' the average electric current as:
$I_{avg}=\frac{\Delta q}{\Delta t}$
If amount of charge passing through the area every second remains constant, then $\Delta q$ is constant so the current is constant.
If the charge flow varies with time, then we can define the instantaneous current by taking the limit as $\Delta t\rightarrow0$ to get:
$i=\frac{dq}{dt}$
The concept of the 'instantaneous' current through a circuit is further complicated when we consider the current through a capacitor element, which requires the introduction of a displacement current to account for changes in electric displacement, the full description of which is given by Maxwell's equations. 
A: We define current as change in charge per time through an area $A$, not in and out of a volume. If the charge is moving through this area you get a current as expected because you have a net flow. If equally much charge passes through from both sides, the current is zero and the net charge is not moving.
A: Realise that, in case of electric current, it is the free electron that moves while the nucleus stays fixed. Hence, when we say charge, in this case, we actually refer to the free electrons that are moving inside the wire.
We do not consider the positive nuclear charges. Obviously the net charge is $0$ across any cross section if we consider the positive charges; else there will be a non zero electric field inside the wire and the current flow will be in haphazard directions.
And, to add more, current is the flow of charges through a conducting wire. Since only the free electrons are mobile while the nuclei are not (from the frame of the observer standing outside the wire on the ground), the flowing charges are the electrons only and that is what matters most, as I have mentioned earlier.
A: Current is not related to the charge "in an infinitesimally thin cross-section of a wire". Electrical current is the amount of charge that passes across that cross-section per unit of time.
If the current is constant, the last expression you show works: 
$ I = \displaystyle\frac{q}{t} $,
If the current varies, your first expression is appropriate.
