Topics in calculus

Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem

Differentiation

Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives

Integration

Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution

In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient 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, for all in some open set containing the number , with ≠ ; and, such that and both exist; then, exists as well:

Examples

The derivative of is:

In the example above, the choices

were made. Analogously, the derivative of (when ≠ 0) is:

For more information regarding the derivatives of trigonometric functions, see: derivative.

Another example is:

whereas and , and and .

The derivative of is determined as follows:

Proofs

From Newton's difference quotient

Suppose

:where ≠ 0 and and are differentiable.

From the product rule

Suppose

The rest is simple algebra to make the only term on the left hand side of the equation and to remove from the right side of the equation.

Alternatively, we can just apply the product rule directly, without having to use substitution:

Followed by using the chain rule to differentiate :

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 old 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 variables 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.

Mnemonic

It is often memorized as a rhyme type song. "Lo-dee-hi, hi-dee-lo, draw the line and square below"; Lo being the denominator, Hi being the numerator and "dee" being the derivative. Another variation to this mnemonic is given when the quotient is written with the numerator as Hi the denominator as Ho: "Ho-dee-Hi minus Hi-dee-Ho all over Ho-Ho." A third variation is "Low-dee-high minus high-dee-low, all over the square of what's below". A fourth variation, similar to the first is "top d bottom minus bottom d top over bottom squared". Here top is the numerator, bottom the denominator, and d meaning derivative. Yet another variation is "Lo-dee-Hi minus Hi-dee-Lo, square the bottom and away we go" where "Hi" is the numerator and "dee" is the derivative.

Often, though, people will remember that the quotient rule is just like the product rule except for two things; the whole thing is divided by the square of the function from the denominator and the addition in the product rule is now changed to subtraction. Most people remember that the product rule is "the derivative of one function times the original of the other plus vice versa," and since it is addition, the order of the addends does not matter. However, with the quotient rule, subtraction now makes that order matter very much, and remembering that order is usually the "sticking point" for most people. Thinking of the numerator function as coming first, with the denominator function following it, yields the following mnemonic: derivative of the numerator times original of the denominator minus original of the numerator times derivative of the denominator, or derivative times original minus original times derivative, or do - od, or dood.

See also

• Product rule
• Chain rule
• Reciprocal rule
• Difference quotient

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
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The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite
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In mathematics, the limit of a function is a fundamental concept in analysis. Informally, a function f(x) has a limit L at a point p if the value of f(x) can be made as close to L as desired, by making x close enough to
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Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions^[1]).
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The term tensor has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function.
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secant joining the endpoints of the interval [a, b] is parallel to the tangent at c.]] In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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In calculus, the product rule also called Leibniz's law (see derivation), governs the differentiation of products of differentiable functions.

It may be stated thus:

or in the Leibniz notation thus:

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In calculus, the chain rule is a formula for the derivative of the composite of two functions.

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
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In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable.
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Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point.
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In differential calculus, related rates problems involve finding the rate at which a quantity is changing by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time.
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The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are differentiable functions, with respect to x, from the real numbers, and c is a real number.
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INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
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See the following pages for lists of integrals:

• List of integrals of rational functions
• List of integrals of irrational functions
• List of integrals of trigonometric functions
• List of integrals of inverse trigonometric functions

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In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
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Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.

It makes use of the so-called "representative cylinder".
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In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians.
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In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing the radical expressions:

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Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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In mathematics, a quotient is the end result of a division problem. For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor. The quotient can also be expressed as the number of times the divisor divides into the dividend.
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trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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In mathematics (more precisely in differential calculus), the term total derivative has a number of closely related meanings.

• The total derivative of a function of several variables, with respect to one of its variables, is, in contrast to the partial derivative, a

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