The differential on a car allows the wheels on the inside of a turn to travel a smaller distance than the wheels on the outside of the turn. If you can't understand why this is important, I pity you.

When one wheel moves it applies pressure on the other wheel in the opposite direction. If you ever have your car up on blocks turn one of the wheels, you'll see the other one turn in the other direction.

The above writeup is indeed correct, but I should point out two more things here.

The differential is sometimes referred to as the rear end, and transfers power from the drive shaft to the rear axles.
A four wheel drive (4WD) usually uses 3 differentials! It has 2 sets of powered wheels (front and rear), and each pair needs a differential to split power on turns.

A third differential is used to divide power between the front pair and the rear pair (when the vehicle is sliding, keeping the wheels rolling at road speed helps prevent the tyres from skidding).

But if the vehicle is stuck with any wheel on a loose surface (e.g. if a wheel is spinning on sand), all power will go to that wheel, and you'll never get out! So this differential can be locked, to allow power to get to the other pair of wheels even while one pair is spinning.

"Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in the narrowest limits, no limits inhere
What joy to discern the minute in infinity!
The vast to perceive in the small, what Divinity!"
- Jakob Bernoulli

As noted by Varberg, Rigdon, and Purcell in Calculus Eigth Edition, the calculus is the study, or analysis, of limits. While algebra focuses on operating with the unknown, and geometry defines the relationships of space, calculus defines the imperceptibly small and the infinitely big. So begins our long journey through this new analysis, central to which is the elusive creature known as the differential.

The originators of the calculus are many, but first credit is usually given to Newton - that brilliant scientist and mathematician who wrote the Principia Mathematica and gave us the concept of a derivative and an integral, and the first methods of dealing with infinitely small changes at instants of time. In addition, it is Newton's methods of the calculus that have been rigorously established by the likes of Cauchy, Riemann, and Lagrange.

However, it is from the work of Leibniz that the concept of the differential itself stems. Leibniz worked on the development of the calculus at the same time as Newton, and contributed greatly to mathematical analysis while also writing several notable philosophical works. In particular, Leibniz used this notation for the derivative:

```dy    d²y
-- or ---
dx    dx²
```

In this case, these are symbols for the first and second derivatives of y with respect to x. However, the key to the differential lies in the reason for using "dx" or "dy". Remember that an average rate of change can be represented this way:

```Δy
--
Δx
```

The beauty of Leibniz's notation lies in recognizing that a derivative is simply a ratio of infinitely small changes. Thus rather than writing "Δx", we simply write "dx", with the d showing an infinitely small change. This infinitely small change is called the differential with respect to x. So, what does this give us? According to the theorems of the calculus, we cannot simply multiply and divide by differentials. We must apply the two new operations of calculus: differentiation and integration. Differentiation, or "taking the derivative", takes an expression and turns one of the variables into a differential, modifying the expression accordingly. For example, when we differentiate over x,

```4x7
```

becomes

```28 x6dx.
```

In reverse, the integral is the infinite sum of differentials. Integrating some function multiplied by its differential reverses the differentiation and gives the antiderivative of that function. This is the heart of the Fundamental Theorem of Calculus. To emphasize: an infinite sum of differentials yields the expression that the differential is obtained from.

However, it is possible to manipulate differentials that are extant in an equation -- you cannot add them in at random, but if there is a dx on one side of an equation you can divide by it to move it to the other side. This technique is known as separation of variables, and is used in many types of differential equations. Unfortunately, it is not entirely legal to do this. Although it works, and whole fields of study are based on the ability to manipulate the differential, the rigorous proof of the calculus does not allow for differentials to be manipulated in this way. It is, after a fashion, cheating.

The puzzle of the differential will intrigue many for years to come.

#### Sources

An automobile's differential gear (often abbreviated "diff", and most commonly referred to as the "differential") is a device which, potentially among other things, allows the wheels to be driven at different speeds. In a turn, the wheels (all of them, but we are currently interested in those which are driven, meaning that power is applied to the road or other surface through them) travel different distances in the same period of time, so this is important. Generally speaking, every car needs a differential of some sort. The differential is sometimes called a "rear end" (as in, nine inch rear end.) In a front wheel drive (FWD) car, the differential is built into the transmission. On a rear wheel drive (RWD) car with a front-mounted engine, it is separate; Either a distinct unit in cars with independent rear suspension (IRS) or built into the middle of the live axle on cars without. On mid-engine and rear engine cars, it is built into the transaxle.

It is not strictly true that every car needs a differential. In some forms of racing (primarily drag racing) it is advantageous to have a "locked differential", also known as a spool because both wheels turn the same, and only the same. Technically, a spool is not a differential (though you can make one by welding some types of differentials together) but a transfer case, as is usually used in the center of four wheel drive (4WD/4x4) vehicles -- but more on that later.

VARIANTS

Open

The most commonly employed type of differential (because it is simple and thus inexpensive and durable) is the open differential, which simply consists of a series of gears. Provided that they have minimal friction and deflection, open differentials are an effective method of avoiding wheel hop of the driven wheels in turns. They do have some serious disadvantages, however; If one wheel is off the ground, then it will spin freely, and the other wheel will not apply any meaningful amount of power to the ground. Also, they tend to slip in wet conditions, which can be hazardous; When they begin to slip, they tend to continue to slip. Because the slipping wheel gets all the power they are not useful for off-road use, either.

There are two solutions to this problem; the limited slip differential (or LSD) and the locking differential. Both types come in automatic and non-automatic versions.

Limited Slip (LSD)

Limited slip differentials further come in four types; viscous, clutch-type or clutch pack, or mechanical. "Posi-traction" is a generic GM-used name for limited slip differential, and can apply to a variety of methods but in the case of GM usually means a clutch-type limited slip.

Viscous LSD works on the same principle as an automatic transmission, which is that the fluid inside of it heats up as it shears, changing its viscosity. When a wheel slips, power is transferred to the other wheel, so power is applied smoothly. There is very little loss under power in such a system, but there is some loss of power during slip. Viscous limited slip units effectively never wear out unless a seal fails -- the fluid could theoretically be burned up, but the engine would probably die first. Some Japanese sports cars (especially from Nissan) have been known to feature a viscous LSD as an option.

Clutch-type LSD uses clutches (high-friction pads under tension) to allow a specified amount of slip. They come in 1, 1.5, and 2 way variants, which number tells you whether they slip only under power, or also during engine braking. They are the favorites of drift racers because of their consistent output and tunability but the clutches wear out and must be replaced periodically.

There are also electronic clutch-type limited slip differentials. These are analogous to antilock brakes (ABS) but for the differential's slip ratio. Eaton makes a limited slip which accepts a pulse width modulated (PWM) signal which determines the amount of engagement given to a particular side of the differential. This type of differential is ideal for traction control applications. Subaru uses an electronically-controlled differential known as the "VCD" in many of its higher-end vehicles to control the split of power between the front and rear of the car.

Mechanical LSD such as the torsen differential (torque-sensing) uses only gears. In the torsen diff's case, it is an open differential with a plus; there are gears which, when they bind, will transfer power to the wheel with traction. By changing the ratio of these added gears one can set the ratio of transferred power, so if they are geared to 5:1, the wheel with traction will have five times the power of the wheel without. However, if a wheel loses too much traction (The larger this ratio is, the less power it can afford to lose) then the torsen diff will not transfer any power, because N times 0 is always 0. Torsen diffs are used in the HMMWV/Hummer and in Quattro-model Audis and 4Motion Volkswagens.

Locking

There are basically two categories of locking differentials, automatic and manual or externally actuated. Most locking diffs use a solenoid or a pneumatic actuator, and are manually controlled. This is most popular for off road use, particularly on dirt. You can lock the halves of the differential together so that you know that you will always apply the same amount of power at both sides.

Tractech, Inc. designed the "Detroit Locker" which is locked most of the time, but in a turn the gears unmesh, which can cause a loud chattering sound that sounds like your diff is about to explode during cornering. They also now have a Detroit Gearless Locker which uses clutch tension to engage or disengage the sides of the differential, though the power is not actually transmitted to the clutch as it is in a limited slip system.

Spool

Sort of an anti-differential, a spool can be a made piece which is installed, or created by welding the spider gears in an ordinary open differential. It forces both wheels to move in the same way. Obviously this means it is not much use in cornering, but it is used in drag racing, both on the street and the strip.

ALL (AND FOUR) WHEEL DRIVE

As for systems with more than one set of drive wheels, the situation becomes more complicated. There are two types of systems in which all four wheels of an automobile are driven.

Four wheel drive (4WD) systems are usually part-time, meaning that they must be engaged manually, either by shifting a lever or by pushing a button. Sometimes the hubs of the wheels also must be engaged, either manually (by spinning) or electrically. Four wheel drive systems have a transmission intended for rear wheel drive use. Most of them have a transfer case in the center, which differs from a differential in that it drives the output shafts (which go to the front and rear differentials) at the same rate, or at different, specified rates. Sometimes these can be replaced with a differential, which means the front and rear wheels can be driven at different rates.

AWD, or all wheel drive, utilizes a transmission which is designed to drive two sets of wheels, with a differential built in. The fast majority of these are modified versions of front wheel drive transmissions. As is typical of modern unibody cars, the engine is mounted transversely and the transmission is closely mated to it, using as little space as possible. A drive shaft goes to the rear differential, and there is a differential for the front wheels built in.

Typically, an all wheel drive system features a limited slip differential in the rear (usually clutch-type or viscous) and an open differential in both the center (controlling front/back balance) and the front. It is generally possible to replace all differentials with superior models, however. The aforementioned (see above) Subaru "VCD" which is electronically controlled works in conjunction with the ECU (Engine Control Unit) to maintain maximum traction. In vehicles with limited slips fore and aft and a variable or limited slip diff in the center, this means that the individual wheel with traction receives the majority of the power. When combined with ABS this gives unparalleled traction control as power can be delivered to any given wheel while braking force is modulated at all others.

References:

1. Nice, Karim. How Differentials Work. about.com, 2003. (http://auto.howstuffworks.com/differential8.htm)
2. Eaton Automotive. Electronic Limited Slip Differentials (http://www.automotive.eaton.com/product/traction_stability/limitedslip.html)

Thanks to Transitional Man.

Audi's have an automatically-disengaging manual locking differentials. A locking rear differential simply locks the rear axle together so the tires spin with the same power. This is beneficial if the vehicle is stuck in mud or snow because with all wheel drive, this means you have 3 out of the 4 tires gripping, the fourth is the tire not spinning on the front differential.

Beginning in the mid-1980's, Audi's had a toggle switch to turn the lock the rear differential from the console. The problem was that people would forget to turn it off when they escaped the trouble and then the locked rear differential really tears up the tires and differential at any speed above 15 mph. Beginning in the 1990's, Audi modified the system so that it automatically disengages the lock and returns to normal differntial mode after 25 mph.

Dif`fer*en"tial (?), a. [Cf. F. diff'erentiel.]

1.

Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate.

For whom he produced differential favors. Motley.

2. Math.

Of or pertaining to a differential, or to differentials.

3. Mech.

Relating to differences of motion or leverage; producing effects by such differences; said of mechanism.

Differential calculus. Math. See under Calculus. -- Differential coefficient, the limit of the ratio of the increment of a function of a variable to the increment of the variable itself, when these increments are made indefinitely small. -- Differential coupling, a form of slip coupling used in light machinery to regulate at pleasure the velocity of the connected shaft. -- Differential duties Polit. Econ., duties which are not imposed equally upon the same products imported from different countries. -- Differential galvanometer Elec., a galvanometer having two coils or circuits, usually equal, through which currents passing in opposite directions are measured by the difference of their effect upon the needle. -- Differential gearing, a train of toothed wheels, usually an epicyclic train, so arranged as to constitute a differential motion. -- Differential motion, a mechanism in which a simple differential combination produces such a change of motion or force as would, with ordinary compound arrangements, require a considerable train of parts. It is used for overcoming great resistance or producing very slow or very rapid motion. -- Differential pulley. Mach. (a) A portable hoisting apparatus, the same in principle as the differential windlass. (b) A hoisting pulley to which power is applied through a differential gearing. -- Differential screw, a compound screw by which a motion is produced equal to the difference of the motions of the component screws. -- Differential thermometer, a thermometer usually with a U-shaped tube terminating in two air bulbs, and containing a colored liquid, used for indicating the difference between the temperatures to which the two bulbs are exposed, by the change of position of the colored fluid, in consequence of the different expansions of the air in the bulbs. A graduated scale is attached to one leg of the tube. -- Differential windlass, ∨ Chinese windlass, a windlass whose barrel has two parts of different diameters. The hoisting rope winds upon one part as it unwinds from the other, and a pulley sustaining the weight to be lifted hangs in the bight of the rope. It is an ancient example of a differential motion.

Dif`fer*en"tial, n.

1. Math.

An increment, usually an indefinitely small one, which is given to a variable quantity.

⇒ According to the more modern writers upon the differential and integral calculus, if two or more quantities are dependent on each other, and subject to increments of value, their differentials need not be small, but are any quantities whose ratios to each other are the limits to which the ratios of the increments approximate, as these increments are reduced nearer and nearer to zero.

2.

A small difference in rates which competing railroad lines, in establishing a common tariff, allow one of their number to make, in order to get a fair share of the business. The lower rate is called a differential rate. Differentials are also sometimes granted to cities.

3. Elec. (a)

One of two coils of conducting wire so related to one another or to a magnet or armature common to both, that one coil produces polar action contrary to that of the other.

(b)

A form of conductor used for dividing and distributing the current to a series of electric lamps so as to maintain equal action in all.

Knight.

Partial differential Math., the differential of a function of two or more variables, when only one of the variables receives an increment. -- Total differential Math., the differential of a function of two or more variables, when each of the variables receives an increment. The total differential of the function is the sum of all the partial differentials.