Why does a capacitor block DC current, but allow AC current to pass through?

The way capacitors and resistors behave is totally
different. While resistors allow a current to flow through them which is proportional to
the voltage drop across the resistor, capacitors oppose a change in voltage across them
by either drawing in or supplying current as they charge or discharge resp. The flow of
current through a capacitor is thus directly proportional to the rate of change of
voltage across it.

This is given by the relation, i = C*
(de/dt) where de/dt is the instantaneous change in
voltage.

As the voltage does not change in the case of DC,
de/dt = 0 and the current that is allowed to pass through by the capacitor is 0. For AC
voltage the voltage changes in a regular manner. Hence here de/dt is not 0 and a current
is allowed to flow through by the capacitor.