Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives Lists of integrals Improper integrals Integration by:
parts,
disks,
cylindrical shells,
substitution,
trigonometric substitution In
calculus, a branch of
mathematics, the
derivative is a measurement of how a
function changes when the values of its inputs change. 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 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,
This expression is
Newton's
difference quotient. The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line. Formally, the
derivative of the function
f at
a is the
limit 
of the difference quotient as
h approaches zero, if this limit exists. If the limit exists, then
f is
differentiable at
a. Here
f '(
a) is one of several common notations for the derivative (
see below).
Equivalently, the derivative satisfies the property that
which has the intuitive interpretation (see Figure 1) that the tangent line to
f at
a gives the
best linear approximation 
to
f near
a (i.e., for small
h). This interpretation is the most easy to generalize to other settings (
see below).
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) is the slope of the secant line between (
a,
f(
a)) and (
a +
h,
f(
a +
h)). If
f is a
continuous function, meaning that its graph is an unbroken curve with no gaps, then
Q is a continuous function away from the point
h = 0. If the limit
exists, meaning that there is a way of choosing a value for
Q(0) which makes the graph of
Q a continuous function, then the function
f is differentiable at the point
a, and its derivative at
a equals
Q(0).
In practice, the continuity of the difference quotient
Q(
h) at
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.
Definition via difference quotients The squaring function
f(
x) =
x is differentiable at
x = 3, and its derivative there is 6. This is proven by writing the difference quotient as follows:
From the last expression, we see that the difference quotient equals
6 +
h when
h is not zero and is undefined when
h is zero. (Remember that in the definition of the difference quotient, we divided by
h, so 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) = 2
a.
Example If
y =
f(
x) is differentiable at
a, then
f must also be
continuous at
a. As an example, choose a point
a and let
f be the
step function which returns a value, say 1, for all
x less than
a, and returns a different value, say 10, for all
x greater than or equal to
a.
f cannot have a derivative at
a. If
h is negative, then
a +
h is on the low part of the step, so the secant line from
a to
a +
h will be very steep, and as
h tends to zero the slope tends to infinity. If
h is positive, then
a +
h is on the high part of the step, so the secant line from
a to
a +
h will have slope zero. Consequently the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the
Weierstrass function.
Continuity and differentiability Let
f be a function that has a derivative at every point
a in the
domain of
f. Because every point
a has a derivative, there is a function which sends the point
a to the derivative of
f at
a. This function is written
f'(x) and is called the
derivative function or the
derivative of
f. The derivative of
f collects all the derivatives of
f at all the points in the domain of
f.
Sometimes
f has a derivative at most, but not all, points of its domain. The function whose value at
a equals
f'(a) whenever
f'(a) is defined and is undefined elsewhere is also called the derivative of
f. It is still a function, but its domain is strictly smaller than the domain of
f.
Using this idea, differentiation becomes a function of functions: The derivative is an
operator whose domain is the set of all functions which have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by
D, then
D(
f) is the function
f′(
x). Since
D(
f) is a function, it can be evaluated at a point
a. By the definition of the derivative function,
D(
f)(
a) =
f′(
a).
For comparison, consider the doubling function
f(
x) =2
x;
f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:
The operator
D, however, is not defined on individual numbers. It is only defined on functions:
Because the output of
D is a function, the output of
D can be evaluated at a point. For instance, when
D is applied to the squaring function,
D outputs the doubling function,
which we named
f(
x). This output function can then be evaluated to get
f(1) = 2,
f(2) = 4, and so on.
The derivative as a function Let
f be a differentiable function, and let
f'(x) be its derivative. The derivative of
f'(x) (if it has one) is written
f''(x) and is called the
second derivative of
f. Similarly, the derivative of a second derivative, if it exists, is written
f'''(x) and is called the
third derivative of
f. These repeated derivatives are called
higher-order derivatives.
A function
f need not have a derivative, for example, if it is not continuous. Similarly, even if
f does have a derivative, it may not have a second derivative. For example, let
.
An elementary calculation shows that
f is a differentiable function whose derivative is
.
f'(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have
k derivatives for any non-negative integer
k but no
(k + 1)-order derivative. A function that has
k successive derivatives is called
k times differentiable. If in addition the
kth derivative is continuous, then the function is said to be of
differentiability class C. (This is a stronger condition than having
k derivatives. For an example, see
differentiability class.) A function that has infinitely many derivatives is called
infinitely differentiable or
smooth.
On the real line, every
polynomial function is infinitely differentiable. By standard
differentiation rules, if a polynomial of degree
n is differentiated
n times, then it becomes a
constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.
The derivatives of a function
f at a point
x provide polynomial approximations to that function near
x. For example, if
f is twice differentiable, then
in the sense that
If
f is infinitely differentiable, then this is the beginning of the
Taylor series for
f.
Higher derivatives Main article: Notation for differentiation Notations for differentiation Main article: Leibniz's notation Leibniz's notation One of the most common modern notations for differentiation is due to
Joseph Louis Lagrange and uses the
prime mark, so that the derivative of a function
is denoted
or simply
. Similarly, the second and third derivatives are denoted
and
. Beyond this point, some authors use Roman numerals such as
for the fourth derivative, whereas other authors place the number of derivatives in parentheses:
in this case. The latter notation generalizes to yield the notation
for the
nth derivative of
f.
Lagrange's notation Main article: Newton's notation Newton's notation Euler's notation uses a
differential operator D, which is applied to a function
f to give the first derivative
Df. The second derivative is denoted
Df.
If
y=
f(
x) is a dependent variable, then often the subscript
x is attached to the
D to clarify the independent variable
x. Euler's notation is then written
Dxy or
Dxf(x), although this subscript is often omitted when the variable
x is understood, for instance when this is the only variable present in the expression.
Euler's notation is useful for stating and solving
linear differential equations.
Euler's notation The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. For some examples, see
Derivative (examples). In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using
rules for obtaining derivatives of more complicated functions from simpler ones.
Computing the derivative Main article: Differentiation rules Rules for finding the derivative Main article: Table of derivatives Derivatives of elementary functions The derivative of
is
Here the second term was computed using the chain rule and third using the product rule: the known derivatives of the elementary functions
x were also used.
Example computation See also: vector calculus and multivariable calculus Derivatives in higher dimensions A
vector-valued function y(
t) of a real variable is a function which sends real numbers to vectors in some
vector space R, then
y(
t) can also be written as
. If we assume that the derivative of a vector-valued function retains the
linearity property, then the derivative of
y(
t) must be
, because each of the basis vectors is a constant.
This generalization is useful, for example, if
y(
t) is the position vector of a particle at time
t; then the derivative
y'(
t) is the
velocity vector of the particle at time
t.
Derivatives of vector valued functions Main article: Partial derivative Partial derivatives Main article: Directional derivative Directional derivatives Main article: Total derivative The total derivative, the Jacobian, and the differential Main article: Derivative (generalizations) Generalizations Online books Automatic differentiation Differentiability class Differintegral Linearization Numerical differentiation Techniques for differentiation
Georg Wilhelm Friedrich Hegel's
Elements of the Philosophy of Right (
Grundlinien der Philosophie des Rechts) was published in
1820, though the book's original title page dates it to
1821. This work is Hegel's most mature statement of his
legal,
moral,
social and
political philosophy and is an expansion upon concepts only briefly dealt with in the
Encyclopedia of the Philosophical Sciences, published in 1817 (and again in 1827 and 1830).
Law provides for Hegel the cornerstone of the modern state. As such, he ironically criticized
Karl Ludwig von Haller's
The Restauration of the Science of the State, in which the latter claimed that law was superficial, because
natural law and the "right of the most powerful" was sufficient (§258). The absence of law characterized for Hegel
despotism, whether
monarchist or
ochlocracist (§278).
The
Philosophy of Right (as it is usually called) begins with a discussion of the concept of the
free will and demonstrates that the free will can only realize itself in the complicated social context of
property rights and relations,
contracts, moral commitments,
family life, the
economy, the legal system, and the
polity. A person is not truly free, in other words, unless he is a participant in all of these different aspects of the life of the state.
The bulk of the book is devoted to discussing Hegel's three spheres of versions of 'right,' each one larger than the preceding ones and encompassing them. The first 'sphere' is abstract right, in which Hegel discusses the idea of 'non-interference' as a way of respecting others. He deems this insufficient and moves onto the second sphere, morality. Under this, Hegel proposes that humans reflect their own subjectivity of others in order to respect them. The third sphere, ethical life, is Hegel's integration of individual subjective feelings and universal notions of right. Under ethical life, Hegel then launches into a lengthy discussion about family, civil society, and the state.
Hegel also demonstrates that the state itself is subsumed under the higher totality of world history, in which individual states arise, conflict with each other, and eventually fall. The course of history is apparently toward the ever-increasing actualization of
freedom; each successive historical epoch corrects certain failures of the earlier ones, but Hegel does not seem to have figured out--and he admits as much--how the modern state can solve the problems of
poverty and
class division, something his successor
Karl Marx would attempt to solve. At the end of his
Lectures on the Philosophy of History, Hegel leaves open the possibility that history has yet to accomplish certain tasks related to the inner organization of the state.
Ž (
minuscule: ž) is: the 25th letter of the
Slovenian alphabet, the 30th letter of the
Serbian; the 42nd letter of the
Czech; the 19th letter of the
Estonian; the 33rd letter of the
Latvian; the 32nd letter of the
Lithuanian; the 46th letter of
Slovak; the 13th letter of the
Turkmen alphabet and the 29th letter of the
Sami alphabet. In addition, the character is also used as a transliteration of Cyrillic "Ж" in
Macedonian (in 31st position) and in
Finnish and occasionally in
Bulgarian (with no special running order of Roman characters). It denotes a similar
phoneme to the Portuguese and French
J. Apart from Estonian and Turkmen, this is the final letter of the other alphabets. In English it is pronounced like the g in genre.
It is formed by addition of a
caron to the standard
Z letter used in most alphabets based on
Latin script and it is pronounced (
IPA)
/ʒ/ in all named alphabets except for Turkmen alphabet, where it is pronounced (IPA)
/ʤ/. Like its
Latin counterpart, the
Cyrillic alphabet has a specific symbol:
Ж.
The
córdoba (
ISO 4217 code: NIO) is the
currency of
Nicaragua. It is divided into 100
centavos.
History Coins In 1912, coins were introduced in denominations of ½, 1, 5, 10, 25 and 50 centavos and 1 córdoba. The ½ and 1 centavo were minted in bronze, the 5 centavos in cupro-nickel and the higher denominations in silver. The 1 córdoba was only minted in 1912, whilst ½ centavo production ceased in 1937.
In 1939, cupro-nickel replaced silver on the 10, 25 and 50 centavos. In 1943, a single year issue of brass 1, 5, 10 and 25 centavos was made. These were the last 1 centavo coins. In 1972, cupro-nickel 1 córdoba coins were issued, followed, in 1974, by aluminium 5 and 10 centavos.
A new series of coins, featuring a portrait of
Augusto César Sandino, was introduced in 1981, consisting of aluminium 5 and 10 centavos, nickel-clad-steel 25 centavos and cupro-nickel 50 centavos, 1 and 5 córdobas. Nickel clad steel replaced cupro nickel between 1983 and 1984. In 1987, the final coins of the first córdoba were issued, featuring Sandino's characteristic hat. Aluminium 5, 10 and 25 centavos and aluminium-bronze 50 centavos, 1 and 5 córdobas were issued, together with aluminium 500 córdobas.
First córdoba No coins were issued for this currency.
Second córdoba In 1994, coins were issued in denominations of 5, 10, 25 and 50 centavos. All were minted in chrome-plated steel. In 1997, nickel-clad-steel 50 centavos, 1 and 5 córdobas were introduced, followed by copper-plated-steel 5 centavos and brass-plated-steel 10 and 25 centavos in 2002.
All current coins have the
coat of arms of the country on the obverse and the numeral of the denomination on the reverse.
Third córdoba (córdoba oro) "EN DIOS CONFIAMOS" - In God we trust
Remarks First córdoba The second córdoba was only issued in banknote form. Notes (dated 1985) were issued in 1988 in denominations of 10, 20, 50, 100, 500 and 1000 córdobas together with undated 5000 córdobas.In 1989, notes for 10,000, 20,000, 50,000 and 100,000 córdobas were introduced, followed the next year by 200,000, 500,000, 1 million, 5 million and 10 million córdobas notes.
Second córdoba In 1991, notes were introduced for 1, 5, 10 and 25 centavos, ½, 1, 5, 10, 20, 50 and 100 córdobas. The notes below 1 córdoba were replaced by coins in 1994, with 5 córdobas notes also being replaced in 1997. 500 córdobas notes were introduced in 2002.
Famous people from Nicaragua's history are depicted on the obverses of the current banknotes. The reverses depict landmarks or natural habitats in the country.
Historical exchange rates Economy of Nicaragua
Raúl Javier Ibáñez (
IPA:
/eˈbaɲez/) (born
June 2,
1972 in
New York, NY) is a left fielder in
Major League Baseball who plays with the
Seattle Mariners. He bats left handed and throws right handed
From
1996 through
2006, Ibánez has posted a .283
average with 138
home runs and 579
RBI in 1,073
games.
Ibáñez attended
Miami-Dade Community College and was an all-conference selection. He was selected by the Seattle Mariners in the 36th round of the
1992 Free Agent Draft. Ibáñez played as a
catcher and outfielder at three levels in the Mariners organization, including his major league first game in 1996. Selected an All-Star in
1994-
1995, he was honored by
Baseball America as its Class-A All-Star catcher for 1995.
A very talented player, Ibáñez was used primarily by the Mariners as a
pinch-hitter and reserve catcher-outfielder. He was signed by the Kansas City Royals as a
free agent before the
2001 season.
Ibáñez finished his first year for the Royals with a .280 average and 54 RBI. He teamed with
Mike Sweeney and
Carlos Beltrán to become just the second trio in major league history to homer in the same three consecutive games. The only other teammates to accomplish the same feat were
Dante Bichette,
Vinny Castilla and
Larry Walker with
Colorado in 1995.
Ibáñez enjoyed his most productive season in
2002, when he collected career-highs in home runs (24), RBI (103),
doubles (37) and
triples (6), and hit .294 with 70
runs. In
2003 he batted 18 homers with 90 RBI and hit .294 for the second straight season (179-for-608); led the team with 33 doubles, and was second with 157 games played, 95 runs and 90 RBI. He also posted career-highs in games,
at-bats (608), runs, hits,
walks (49) and
stolen bases (10).
After three seasons with the Royals, Ibánez returned to Seattle in 2004 and hit a career-high .304 average. He also collected a 24-game go-ahead RBI string, matched a club record by reaching base 11 consecutive times, set a career-high and a club record and matched the
American League record with six hits, and joined
Ichiro Suzuki as only Mariners ever with two five-hit games in one season.
With
Edgar Martinez retired, Ibáñez moved to the DH spot before the
2005 season. In
2005 Ibánez hit .280 with 20 home runs and 89 RBI.
Since the Mariners acquired switch designated hitter
Carl Everett from
Chicago White Sox during the 2005 off season, Ibánez has been moved back to where he played before, namely the left field for 2006 season from the beginning.
Ibanez finished the 2006 season with a .289 batting average, and set career highs with 33 home runs and 123 runs batted in.
Werewolves of London is his at-bat song.
Seattle Mariners (
1996–
2000)
Kansas City Royals (
2001–
2003)
Seattle Mariners (
2004-present)
Similarities between different religious mythologies
Modern-day opposition
