Question: I understand that the differential (or "diff") is the thing that allows the drive shaft to transfer its rotational velocity to the front or rear axle in order to move the car. As far as I know, the diff is the part that also allows each wheel to turn independently of each other, so as not to drag the outside tire going round a corner, which would increase tire wear and decrease fuel consumption.

* ***How exactly does a differential work?***
* ***How does the diff enable each wheel to turn at different speeds?***
* ***How are some cars able to dynamically adjust the amount of power to each wheel (in more high tech systems)?***

Answer: 
What is a Differential ? 

* A differential’s job is to compensate for differences. Specifically
the differences in wheel speed when turning. For instance, imagine
taking a corner. Your inside wheel has a shorter distance to travel
than the outside wheel as you go around the corner. That means that
your outside wheel has to turn faster to keep pace with the inside
wheel.
* The differential allows the wheels to turn at different speeds while
still providing power. Without a differential, the tires would scrub
and hop along the pavement during turns.

Working Principle:

* I found an excellent online source [here](http://mechanicalmania.blogspot.com/2011/07/construction-working-of-differential.html)
* Torque is supplied from the engine, via the transmission, to a drive
shaft, which runs to the final drive unit that contains the
differential.
* A spiral bevel pinion gear takes its drive from the end of the
propeller shaft, and is encased within the housing of the final drive
unit. This meshes with the large spiral bevel ring gear, known as the
crown wheel.
* The crown wheel and pinion may mesh in hypoid orientation, not shown.
The crown wheel gear is attached to the differential carrier or cage,
which contains the 'sun' and 'planet' wheels or gears, which are a
cluster of four opposed bevel gears in perpendicular plane, so each
bevel gear meshes with two neighbours, and rotates counter to the
third, that it faces and does not mesh with.
* The two sun wheel gears are aligned on the same axis as the crown
wheel gear, and drive the axle half shafts connected to the vehicle's
driven wheels. The other two planet gears are aligned on a
perpendicular axis which changes orientation with the ring gear's
rotation.

Basically no matter which wheel experiences load the planetary gears will corresponding to them will compensate for that.

The ability of a diff to send various amounts of power on demand to individual wheel is called a Torque vectoring.

They generally have a electric actuator attached to the diff assembly to assist with the torque distribution

For more advanced info on torque vectoring , [This](https://www.youtube.com/watch?v=qwwFZAbYGW0) video will be helpful.
[![enter image description here](https://i.stack.imgur.com/DqLIi.gif)](https://i.stack.imgur.com/DqLIi.gif)

Additional info on torque vectoring can be found in this good question [How does torque diff work](https://mechanics.stackexchange.com/questions/23103/lexus-rcf-torque-vectoring-differential-how-does-it-work)
<https://mechanics.stackexchange.com/a/25370/12697> Covers the physical arrangment of gears in a basic differential. The key to understanding a differential is to understand how these gears behave under different scenarios.

The gearing arrangement means that relative to the crown wheel the two half shafts are constrained to turn by equal and opposite ammounts.

Looking from outside the differential this means that the average speed of the two half shafts must be equal to the speed at which the crown wheel turns. So when the engine turns the propshaft at least one of the wheels must turn.

We also see that if the differential is frictionless then the torque delivered to the two wheels must be the same since any difference in torque would cause the gears inside the differential to change speed.

If both the drive wheels have good grip then the basic differential works great at transferring power to the road while allowing the drive wheels to turn at different speeds.

The same principle can be extended to more wheels, for example a 4 wheel drive vehicle will usually have three differentials, one in the center for differences between front and back and then front and back differentials for differences in speed between left and right of each end.

If one wheel loses grip then the wheel that loses grip will spin freely since the torque delivered to the wheels is the same very little power will be delivered to the wheel(s) that still has grip.

So in performance cars systems are built to detect undesirablly high levels of slip and limit them either through the cars brakes or through extra mechnisms in the differntial. There are also more complex types of differential that have different torque VS speed characteristics.

Losing traction on your 4 wheel drive vehicle when one wheel slips kind of defeats the object of having 4 wheel drive. Old fasioned off-roaders often used crude mechnisms to lock up the differentials when driving off road. More modern off roaders often have fancier systems similar to those on performance cars.