The differential is the transfer point, in most automobiles, where the power is delivered from the transmission (manual or automatic) to the wheels. It is also what allows the wheels to turn at different speeds.
The wheels of your vehicle turn at different speeds, particularly when driving on a winding road or turning a corner. The wheels on the inside of a turn travel a shorter distance than the outside wheels. Speed is equal to the distance traveled, divided by the amount of time it takes to cover that distance, so the wheels that travel a shorter distance travel at a lower speed and the front wheels will travel a different distance than the rear wheels.
The non-driven wheels on your car (the ones that don't get power from the engine), are the front wheels on a rear-wheel drive car or the back wheels on a front-wheel drive car don't have a problem turning at different speeds. They're not directly connected to each other, so they turn independently. The driven wheels are connected, to allow a single engine and transmission can turn both wheels. Without a differential, the wheels would have to be locked together, forcing them to turn at the same speed. That would make turning difficult and much harder on the car. To allow your car to be able to turn, one of the tires would have to slip. Modern tires generate a significant amount of "roud-grip" and concrete roads tend to provide excellent, so a huge amount of force is required to make a tire slip. That energy would have to be transmitted through the axle from one wheel to another, putting a tremendous strain on the axle components.
All modern cars and trucks have a differential, as do many full-time, four-wheel-drive vehicles. These all-wheel-drive vehicles need a differential connecting each set of drive wheels, and they also need one between the front and the rear wheels, because the front wheels travel a different distance through a turn than the back wheels.
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