What change does $dQ$ represent in definition of current $i$ The definition of current $i$ is
$$i=\frac{dQ}{dt}.$$
According to calculus whenever we write one variable as a derivative  of another variable that simply means we are trying to calculate the rate of change of former variable with respect to the latter but in the definition of current $dQ$ doesn’t seem to represent any change rather it is amount of charge passing through a particular area but since we are writing $Q$ as a derivative of time that means we are trying to calculate the rate of change of $Q$ with respect to time but actually this is not what we desire to calculate then why we are writing $Q$ as the derivative of time although $dQ$ does not represent any change.
The same argument applies to definition of rates of flow (for example water).
I may be getting wrong somewhere since I am a newbie to current electricity so please make me correct where I am getting wrong so that I can understand why we are write $Q$ as a derivative of time.
 A: $Q(t)$ can be regarded as the total charge which has flown through a cross-sectional area and perpendicular to it from some time $t=t_{0}$ to $t=t$, where $t_0<t$. In general, $t_{0}$ would be the time when you turn on the current. So, while you can think of $dQ$ as being the differential amount of charge flowing through the cross-section in differential time $dt$, you can also think of $dQ$ as being the change in the total charge which has flown through the cross-section, which occurs in time $dt$. Therefore, $\frac{dQ}{dt}$ is the rate of change of "the total charge which has flown through the cross section" with respect to time.
A: It is a good question.
I prefer the concept of density of current $\mathbf j = \rho \mathbf v$, where $\rho$ is the density and $\mathbf v$ the velocity of charges. The current $\mathbf I = \mathbf jS$, where $S$ is the cross section of the conductor. Density of charges and currents are the units for the sources used in the Maxwell equations.
The same for fluid flow: $\mathbf Q = \mu \mathbf vS$, where $\mu$ is the density and $\mathbf v$ the velocity of the fluid.
A: In defining current the dQ is the small quantity of charge crossing a given cross section of the conductor in a corresponding short time dt.  The same concept applies to fluid flow.
A: Instead of tiny $dQ$ charges moving through a conductor, it is simple to think of $dQ$ as the amount of charge that's present in one point on the conductor, and the current is the rate in which that amount changes. Same with other types of flow, filling a pool with a hose we can define the amount of water in the pool, and the flow in a similar way. When we then think of it differentially, for an infinitesimal point on the conductor/hose, it allows us to define a current/flow for any point we pick on the conductor/hose.
A: I think the main doubt you are having is that you are assuming charge to be a continuum like water. However charge is not continuum. It is quantisized.
Consider you have a simple circuit with one battery. Now , the net charge of the circuit is zero. Consider a charge $dQ$ coming out of the negative terminal of the battery.
Now this $dQ$ will rush towards the positive terminal of the battery through the circuit . Assume it takes $t = \pi$ seconds to complete one trip from the negative terminal to the positive terminal.
Now consider a small area $s$ in the circuit (in the wire/conductor) .
Charge will pass through $s$ only at $t = n.t_0$ seconds. In other times , the charge will be present at other positions in the circuit.
Therefore initially the area $s$ had $0$ charge . But soon at $t = t_0$ seconds , there was charge passing through it.
There was a change of net charge in that area s . This change in charge is represented as $dQ$ in the equation.
Also note that this charge travels at near light speed. So this time $t = \pi$ s is very small and almost negligible in real life. Because of this it appears as if the Charge is flowing like water.
A: This is merely a breakdown of notational convention.
This happens in bookkeeping where the quantity $Q$ of an asset that has passed through an account by time $T$ remains a quantity $Q$ even though it is actually a transactional quantity $ΔQ$ for time interval $ΔT$ and not the integral (balance sheet) asset amount $Q$. This is what is happening with current or fluid flow $dQ$, as the quantity over interval $dt$.
A good illustration is the fluid example in the @claudio-saspinski answer if one calls the integral quantity $V = Qt$ (using dimensional analysis), and if one wanted to find the flow from the quantity you'd take an Algebraic Derivation: $ΔV = ΔQ.t + Q.Δt$, the change of flow at a time instance and the change of time at a flow level, and the normalized change is $ΔV/Δt = ΔQ/Δt\cdot t + Q\cdot Δt/Δt$, whose continuum limit $dV$ is $Q + t.(dQ/dt)$.
When density, cross-sectional area and velocity are all variable the derivation $ΔQ = Δ(μvS)$ is: $Δμ\cdot v\cdot S + μ\cdot Δv\cdot S + \mu\cdot v\cdot ΔS$ arising from volumetric compression, linear acceleration and orifice aperture changes respectively.
