The Derailleur gear system moves the bicycle chain between different gears on both the crank and rear sprocket gearsets. This is achieved by displacing the chain sideways so that it 'jumps' from one gear and engages another in a controlled way. The overall gear ratio is (N_C / N_S) and gives the number of sprocket revolutions per chainwheel revolution.

The advance of the bike per crank revolution is determined by the gear ratio and the diameter of the rear wheel.

A 700 mm rear wheel moves the bike forward by 2.2 m in a single revolution. For this wheel, the distance traveled by the bike for a single crank revolution is:
2.2 (N_C / N_S) m.

The picture shows the rear sprocket mechanism (without the chain). The derailleur mechanism is to the right of the gears. The two guide wheels of this mechanism move in a plane normal to the rear wheel axis and transfer the chain from one sprocket to the next.

From: Apps, "The Bicycle Book," Smithmark (1993)