# Dictionary Definition

derivative adj : resulting from or employing
derivation; "a derivative process"; "a highly derivative prose
style"

### Noun

1 the result of mathematical differentiation; the
instantaneous change of one quantity relative to another; df(x)/dx
[syn: derived
function, differential
coefficient, differential, first
derivative]

2 a financial instrument whose value is based on
another security [syn: derivative
instrument]

3 (linguistics) a word that is derived from
another word; "`electricity' is a derivative of `electric'"

# User Contributed Dictionary

### Pronunciation

### Adjective

derivative- Imitative of the work of someone else
- (copyright law) Referring to a work, such as a translation or adaptation, based on another work that may be subject to copyright restrictions
- Having a value that depends on an underlying asset of variable value
- Lacking originality

### Noun

derivative (plural: derivatives)- Something derived.
- A word that derives from another one.
- A financial instrument whose value depends on the valuation of an underlying asset; such as a warrant, an option etc.
- A chemical derived from another.
- The derived
function of a function.
- The derivative of f:f(x) = x^2 is f':f'(x) = 2x

- The value of this function for a given value of its independent
variable.
- The derivative of f(x) = x^2 at x = 3 is f'(3) = 2*3 = 6.

#### Synonyms

- (in finance): contingent claim
- (in analysis: function): derived function
- spinoff

#### Antonyms

#### Translations

something derived

word that derives from another

financial instrument whose value depends on the
valuation of an underlying instrument

- Czech: derivát
- Finnish: johdannainen
- German: Derivat
- Japanese: デリバティブ (deribathibu)

chemical derived from another

in analysis: function

See derived
function

in analysis: value

- Czech: derivace
- Finnish: derivaatta
- French: dérivée
- German: Ableitung
- Italian: derivata
- Portuguese: derivada
- Slovene: odvod
- Spanish: derivada

- ttbc Spanish: derivativo (1-4)

# Extensive Definition

In calculus, a branch of mathematics, the 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. For example, the derivative of the position or
distance of a car at some point in time is the instantaneous
velocity, or instantaneous speed (respectively), at which that car
is traveling (conversely the integral of the velocity is the
car's position).

A closely related notion is the differential
of a function.

The derivative of a function at a chosen input
value describes the best linear
approximation of the function near that input value. For a
real-valued
function of a single real variable, the derivative at a point
equals the slope of the
tangent
line to the graph
of the function at that point. In higher dimensions, the
derivative of a function at a point is a linear
transformation called the linearization.

The process of finding a derivative is called
differentiation. The
fundamental theorem of calculus states that differentiation is
the reverse process to integration.

## Differentiation and the derivative

Differentiation is a method to compute the rate
at which a quantity, y, changes with respect to the change in
another quantity, x, upon which it is dependent.
This rate of change is called the derivative of y with respect to
x. In more precise language, the dependency of y on x means that y
is a function
of x. If x and y are real numbers,
and if the graph
of y is plotted against x, the derivative measures the slope of this graph at each point.
This functional relationship is often denoted y = f(x), where f
denotes the function.

The simplest case is when y is a linear
function of x, meaning that the graph of y against x is a
straight line. In this case, y = f(x) = m x + c, for real numbers m
and c, and the slope m is given by

- m= =

- y + Δy = f(x+ Δx) = m (x + Δx) + c = m x + c + m Δx = y + mΔx.

This gives an exact value for the slope of a
straight line. If the function f is not linear (i.e. its graph is
not a straight line), however, then the change in y divided by the
change in x varies: differentiation is a method to find an exact
value for this rate of change at any given value of x.

The idea, illustrated by Figures 1-3, is to
compute the rate of change as the limiting
value of the ratio of
the differences Δy / Δx as Δx becomes infinitely small.

In Leibniz's
notation, such an infinitesimal change in x
is denoted by dx, and the derivative of y with respect to x is
written

- \frac \,\!

The most common approach to turn this intuitive
idea into a precise definition uses limits,
but there are other methods, such as non-standard
analysis.

### Definition via difference quotients

Let y=f(x) be a function of x. In classical
geometry, the tangent line at a real number a was the unique line
through the point (a, f(a)) which did not meet the graph of f
transversally,
meaning that the line did not pass straight through the graph. The
derivative of y with respect to x at a is, geometrically, the slope
of the tangent line to the graph of f at a. The slope of the
tangent line is very close to the slope of the line through (a,
f(a)) and a nearby point on the graph, for example (a + h, f(a +
h)). These lines are called secant lines.
A value of h close to zero will give a good approximation to the
slope of the tangent line, and smaller values (in absolute
value) of h will, in general, give better approximations. The slope
of the secant line is the difference between the y values of these
points divided by the difference between the x values, that is,

- \frac.

- f'(a)=\lim_

Equivalently, the derivative satisfies the
property that

- \lim_ = 0,

- f(a+h) \approx f(a) + f'(a)h

Substituting 0 for h in the difference quotient causes division
by zero, so the slope of the tangent line cannot be found
directly. Instead, define Q(h) to be the difference quotient as a
function of h:

- Q(h) = \frac.

In practice, the existence of a continuous
extension of the difference quotient Q(h) to h = 0 is shown by
modifying the numerator to cancel h in the denominator. This
process can be long and tedious for complicated functions, and many
short cuts are commonly used to simplify the process.

### Example

The squaring function f(x) = x² is differentiable
at x = 3, and its derivative there is 6. This is proven by writing
the difference quotient as follows:

- = = = = 6 + h.

Then we get the simplified function in the
limit:

- \lim_ 6 + h = 6 + 0 = 6.

The last expression shows that the difference
quotient equals 6 + h when h is not zero and is undefined when h is
zero. (Remember that because of the definition of the difference
quotient, the difference quotient is always undefined when h is
zero.) However, there is a natural way of filling in a value for
the difference quotient at zero, namely 6. Hence the slope of the
graph of the squaring function at the point (3, 9) is 6, and so its
derivative at x = 3 is f '(3) = 6.

More generally, a similar computation shows that
the derivative of the squaring function at x = a is f '(a) =
2a.

### Continuity and differentiability

## External links

- WIMS Function Calculator makes online calculation of derivatives; this software also enables interactive exercises.
- Online Derivatives Calculator.
- Mathematical Assistant on Web online calculation of derivatives, including explanation of steps in the solution.
- Proving Derivatives from First Principles.
- Practice finding derivatives of randomly generated functions

derivative in Afrikaans: Afgeleide

derivative in Arabic: اشتقاق (رياضيات)

derivative in Bulgarian: Производна

derivative in Catalan: Derivada

derivative in Czech: Derivace

derivative in Danish: Differentialregning

derivative in German: Differentialrechnung

derivative in Estonian: Tuletis
(matemaatika)

derivative in Spanish: Derivada

derivative in Esperanto: Derivaĵo
(matematiko)

derivative in Basque: Deribatu

derivative in Persian: مشتق

derivative in French: Dérivée

derivative in Friulian: Derivade

derivative in Galician: Derivada

derivative in Korean: 미분

derivative in Ido: Derivajo

derivative in Indonesian: Turunan

derivative in Icelandic: Deildun

derivative in Italian: Derivata

derivative in Hebrew: נגזרת

derivative in Lithuanian: Išvestinė

derivative in Lombard: Derivada

derivative in Hungarian:
Differenciálhányados

derivative in Macedonian: Диференцијално
сметање

derivative in Dutch: Afgeleide

derivative in Japanese: 微分法

derivative in Norwegian: Derivasjon

derivative in Polish: Pochodna funkcji

derivative in Portuguese: Derivada

derivative in Romanian: Derivată

derivative in Russian: Производная функции

derivative in Slovak: Derivácia (funkcia)

derivative in Slovenian: Odvod

derivative in Serbian: Извод

derivative in Finnish: Derivaatta

derivative in Swedish: Derivata

derivative in Thai: อนุพันธ์

derivative in Vietnamese: Đạo hàm và vi phân của
hàm số

derivative in Turkish: Türev

derivative in Ukrainian: Похідна

derivative in Contenese: 微分

derivative in Chinese: 导数

# Synonyms, Antonyms and Related Words

accountable, acquired, alleged, ascribable, assignable, attributable, attributed, borrowed, by-product, charged, conjugate, consequent, consequential, copied, credited, derivable from,
derivation, derivational, derived, descendant, development, due, echoic, ensuing, etymologic, explicable, final, following, imitative, imputable, imputed, lexical, lexicographic, lexicologic, lexigraphic, noncreative, nongerminal, nonseminal, obtained, offshoot, onomastic, onomatologic, onomatopoeic, owing, paronymic, paronymous, plagiarized, procured, putative, referable, referred to,
resultant, resulting, sequacious, sequent, sequential, spin-off,
traceable, uncreative, uninventive, unoriginal, unpregnant