Quotient rule

In calculus

Calculus

Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...

, the quotient rule is a method of finding the derivative

Derivative

In calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...

 of a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 that is the quotient

Quotient

In mathematics, a quotient is the result of a division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend....

 of two other functions for which derivatives exist.

If the function one wishes to differentiate, , can be written as
and ≠ , then the rule states that the derivative of is equal to:
Or, more precisely, if all x in some open set

Open set

In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...

 containing the number a satisfy ≠ ; and and both exist; then, exists as well and:


The derivative of is:
In the example above, the choices
were made.

In calculus

Calculus

Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...

, the quotient rule is a method of finding the derivative

Derivative

In calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...

 of a function

Function (mathematics)

In mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...

 that is the quotient

Quotient

In mathematics, a quotient is the result of a division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend....

 of two other functions for which derivatives exist.

If the function one wishes to differentiate, , can be written as
and ≠ , then the rule states that the derivative of is equal to:
Or, more precisely, if all x in some open set

Open set

In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...

 containing the number a satisfy ≠ ; and and both exist; then, exists as well and:

Examples

The derivative of is:
In the example above, the choices
were made. Analogously, the derivative of (when ≠ 0) is:
Another example is:
whereas and , and and .

The derivative of is determined as follows:
This can be checked by using laws of exponents and the power rule:

Limitations

The quotient rule is not useful at points where either the numerator or denominator are not differentiable; it's possible that the quotient may be differentiable at such points. For example, consider the function:
where |x| denotes the absolute value

Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....

 of x. This is of course simply the function f(x) = 1, so it is differentiable everywhere and in particular f'(0) = 0. If we try to use the quotient rule to compute f'(0), however, an undefined value will result, since |x| is nondifferentiable at x = 0.

From Newton's difference quotient

Suppose where and and are differentiable.

We pull out the and combine the fractions in the numerator:

Adding and subtracting in the numerator:

We factor this and multiply the through the numerator:

Now we move the limit through:

By the definition of the difference quotient, the limits in the numerator are derivatives, so we have:

Using the Chain Rule

Consider the identity
Then
Leading to
Multiplying out leads to
Finally, taking a common denominator leaves us with the expected result

By total differentials

An even more elegant proof is a consequence of the law about total differentials, which states that the total differential,

of any function in any set of quantities is decomposable in this way, no matter what the independent variable

Independent variable

The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...

s in a function are (i.e., no matter which variables are taken so that they may not be expressed as functions of other variables). This means that, if N and D are both functions of an independent variable x, and
, then it must be true both that

(*)

and that


But we know that and .

Substituting and setting these two total differentials equal to one another (since they represent limits which we can manipulate), we obtain the equation

which requires that

(#) .

We compute the partials on the right:

;

.

If we substitute them into (#),
which gives us the quotient rule, since, by (*),.

This proof, of course, is just another, more systematic (even if outmoded) way of proving the theorem in terms of limits, and is therefore equivalent to the first proof above - and even reduces to it, if you make the right substitutions in the right places. Students of multivariable calculus will recognize it as one of the chain rules for functions of multiple variables.

External links

• MIT Calculus video lecture on the quotient rule