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How do front and rear derailleurs move the chain between gears?

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An African American family of four, all wearing helmets, rides bicycles together on a tree-lined country trail, with two children in front and two adults following behind. — bicycle Family bicycling on a country road.

A derailleur is a mechanism that shifts gears by physically moving the chain from one set of exposed gears to another. The process works as follows:1

Front Derailleur: This component moves the chain between the front chainwheels (the gears connected to the crank and pedals).21
Rear Derailleur: This component moves the chain between the various sprockets located on the rear wheel hub. It also features a spring-loaded pulley system designed to maintain proper chain tension and take up any slack created during the shifting process.21

Shifting is controlled by the rider using levers, trigger mechanisms, or twist-grip shifters mounted on the handlebars or frame, which are connected to the derailleurs by wires. By moving the chain between different sizes of chainwheels and sprockets, the rider can adjust the gear ratio to match their pedaling cadence to the terrain.21