NationMaster - Encyclopedia: Quotient rule

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem For other uses, see Calculus (disambiguation). ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... 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 (slope) of the curve is equal (parallel) to the average derivative of the section. ...
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives For a non-technical overview of the subject, see Calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... 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. ... In calculus, Taylors 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. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ... The primary operation in differential calculus is finding a derivative. ...
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution This article is about the concept of integrals in calculus. ... 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 List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

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. For other uses, see Calculus (disambiguation). ... For a non-technical overview of the subject, see Calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a quotient is the end result of a division problem. ...

If the function one wishes to differentiate, $f (x)$ , can be written as

$f(x) = frac{g(x)}{h(x)}$

and $h (x)$ ≠ $0$ , then the rule states that the derivative of $g (x) / h (x)$ is equal to:

$frac{d}{dx}f(x) = f'(x) = frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}.$

Or, more precisely, for all $x$ in some open set containing the number $a$ , with $h (a)$ ≠ $0$ ; and, such that $g'(a)$ and $h'(a)$ both exist; then, $f'(a)$ exists as well: In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

$f'(a)=frac{g'(a)h(a) - g(a)h'(a)}{[h(a)]^2}.$

1 Examples
2 Proofs
3 Mnemonic
4 See also

Examples

The derivative of $(4 x - 2) / (x 2 + 1)$ is:

$frac{d}{dx} frac{(4x - 2)}{x^2 + 1}$	$=frac{(x^2 + 1)(4) - (4x - 2)(2x)}{(x^2 + 1)^2}$
	$=frac{(4x^2 + 4) - (8x^2 - 4x)}{(x^2 + 1)^2}$
	$=frac{-4x^2 + 4x + 4}{(x^2 + 1)^2}$

In the example above, the choices

g (x) = 4 x - 2

h (x) = x 2 + 1

were made. Analogously, the derivative of $sin(x) / x 2$ (when $x$ ≠ 0) is:

$frac{cos(x) x^2 - sin(x)2x}{x^4}$

For more information regarding the derivatives of trigonometric functions, see: derivative. In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... For a non-technical overview of the subject, see Calculus. ...

Another example is:

$f(x) = frac{2x^2}{x^3}$

whereas $g (x) = 2 x 2$ and $h (x) = x 3$ , and $g'(x) = 4 x$ and $h'(x) = 3 x 2$ .

The derivative of $f (x)$ is determined as follows:

$f'(x),$	$=frac {left(4x cdot x^3 right) - left(2x^2 cdot 3x^2 right)} {left(x^3right)^2}$
	$=frac{4x^4 - 6x^4}{x^6}$
	$=frac{-2x^4}{x^6}$
	$=-frac{2}{x^2}$

Proofs

From Newton's difference quotient

Suppose

f (x) = g (x) / h (x)

where

h (x)

≠ 0 and

g

and

h

are differentiable.

$f'(x) = lim_{Delta x to 0} frac{f(x + Delta x) - f(x)}{Delta x} = lim_{Delta x to 0} frac{frac{g(x + Delta x)}{h(x + Delta x)} - frac{g(x)}{h(x)}}{Delta x}$

$= lim_{Delta x to 0} frac{1}{Delta x} left( frac{g(x+Delta x)h(x)-g(x)h(x+Delta x)}{h(x)h(x+Delta x)} right)$

$= lim_{Delta x to 0} frac{1}{Delta x} left( frac{(g(x+Delta x)h(x)-g(x)h(x))-(g(x)h(x+Delta x)-g(x)h(x))}{h(x)h(x+Delta x)} right)$

$= lim_{Delta x to 0} frac{1}{Delta x} left( frac{h(x)(g(x+Delta x)-g(x))-g(x)(h(x+Delta x)-h(x))}{h(x)h(x+Delta x)} right)$

$= lim_{Delta x to 0} frac{frac{g(x+Delta x)-g(x)}{Delta x}h(x)-g(x)frac{h(x+Delta x)-h(x)}{Delta x}}{h(x)h(x+Delta x)}$

$= frac{lim_{Delta x to 0} left(frac{g(x+Delta x)-g(x)}{Delta x}right)h(x) - g(x) lim_{Delta x to 0} left(frac{h(x+Delta x)-h(x)}{Delta x}right)}{h(x)h(lim_{Delta x to 0} (x+Delta x))}$

$= frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

From the product rule

Suppose $f(x) = frac{g(x)}{h(x)}$

$g(x) = f(x)h(x) mbox{ } ,$

$g'(x)=f'(x)h(x) + f(x)h'(x)mbox{ } ,$

The rest is simple algebra to make $f'(x)$ the only term on the left hand side of the equation and to remove $f (x)$ from the right side of the equation.

$f'(x)=frac{g'(x) - f(x)h'(x)}{h(x)} = frac{g'(x) - frac{g(x)}{h(x)}cdot h'(x)}{h(x)}$

$f'(x)=frac{g'(x)h(x) - g(x)h'(x)}{left(h(x)right)^2}$

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

$f(x) = frac{g(x)}{h(x)} = g(x) [h(x)]^{-1}$

Followed by using the chain rule to differentiate $h (x) - 1$ :

$f'(x) = g'(x) [h(x)]^{-1} + g(x) (-1) [h(x)]^{-2} h'(x) = frac{g'(x) h(x) - g(x) h'(x)}{[h(x)]^2}$

Using the Chain Rule

Consider the identity

$frac{u}{v}; =; frac{1}{4}left[ left( u+frac{1}{v} right)^{2}-; left( u-frac{1}{v} right)^{2} right]$

Then

$frac{dleft( frac{u}{v} right)}{dx}; =; frac{d}{dx}frac{1}{4}left[ left( u+frac{1}{v} right)^{2}-; left( u-frac{1}{v} right)^{2} right]$

Leading to

$frac{dleft( frac{u}{v} right)}{dx}; =; frac{1}{4}left[ 2left( u+frac{1}{v} right)left( frac{du}{dx}-frac{dv}{v^{2}dx} right)-; 2left( u-frac{1}{v} right)left( frac{du}{dx}+frac{dv}{v^{2}dx} right) right]$

Multiplying out leads to

$frac{dleft( frac{u}{v} right)}{dx}; =; frac{1}{4}left[ frac{4}{v}frac{du}{dx}-frac{4u}{v^{2}}frac{dv}{dx} right]$

Finally, taking a common denominator leaves us with the expected result

$frac{dleft( frac{u}{v} right)}{dx}; =; frac{left[ vfrac{du}{dx}-ufrac{dv}{dx} right]}{v^{2}}$

By total differentials

An even more elegant proof is a consequence of the old law about total differentials, which states that the total differential, In mathematics, a total derivative is a combination of partial derivatives. ...

$dF = frac{partial F}{partial x} dx + frac{partial F}{partial y} dy + frac{partial F}{partial z} dz + ...$

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 $F = N (x) / D (x)$ , then it must be true both that In an experimental design, the independent variable (argument of a function, also called a predictor variable) is the variable that is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ...

(*) $dF = frac{partial F}{partial x}dx$

and that

$dF = frac{partial F}{partial N}dN + frac{partial F}{partial D}dD$ .

But we know that $d N = N'(x) d x$ and $d D = D'(x) d x$ .

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

$frac{partial F}{partial x} dx = frac{partial F}{partial N}N'(x) dx + frac{partial F}{partial D}D'(x) dx$

which requires that

(#) $frac{partial F}{partial x} = frac{partial F}{partial N}N'(x) + frac{partial F}{partial D}D'(x)$ .

We compute the partials on the right:

$frac{partial F}{partial N} = frac{partial (N/D)}{partial N} = frac{1}{D}$ ;

$frac{partial F}{partial D} = frac{partial (N/D)}{partial D} = -frac{N}{D^2}$ .

If we substitute them into (#),

$frac{partial F}{partial x} = frac{N'(x)}{D(x)} - frac{N(x) D'(x)}{D(x)^2}$

$frac{partial F}{partial x} = frac{D(x)N'(x)}{D(x)^2} - frac{N(x) D'(x)}{D(x)^2}$

which gives us the quotient rule, since, by (*),

$frac{dF}{dx} = frac{partial F}{partial x}$ .

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.

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