Capacitor Current–voltage relations conceptual doubt
In a perfect world, the current would be zero yes because once the capacitor "fills up" its charge, it'll behave like an open circuit and thus the current would be zero and your voltage V(t) would be equal to V(t0). Think of that as a taller water tank emptying itself out into another water tank. There will be equilibrium between the two water tanks.



(forgive my drawing, it's not abiding by the laws of physics ◠‿◠ )

However, in the real world, capacitors take time to charge and that voltage is seen as an exponential growth until it plateaus completely. This exponential equation is where you take the derivative. The reason why it's exponential is because there is some resistivity inside the capacitor. This is also known as ESR.