Derivative and List of GMA Network original programming: Difference between pages

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Revision as of 17:07, 2 July 2007

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{{Alternateuses}}

'''[[GMA Network]]'s programs''' include news and current affairs shows, [[documentaries]], [[drama]]s, dubbed foreign serials, entertainment, news and talk shows, game shows, variety shows, musicals, sitcoms, children's shows, fantasy and reality shows.

{{Calculus}}

In [[mathematics]], the '''derivative''' of a [[function (mathematics)|function]] is one of the two central concepts of [[calculus]]. (The other one is the [[antiderivative]], the [[inverse]] of the derivative.)

For the previously aired and defunct shows of the network, please see the [[list of shows previously aired by GMA Network]].

The derivative of a function at a [[point]] measures the rate at which the function's value [[change|changes]] as the function's [[parameter|argument]] [[change]]s. That is, a derivative provides a mathematical formulation of the notion of '''rate of change'''. As it turns out, the derivative is an extremely versatile concept which can be viewed in many different ways. For example, referring to the two-dimensional [[graph of a function|graph]] of ''f'', the derivative can also be regarded as the [[slope]] of the [[tangent]] to the graph at the point ''x''. The slope of this tangent can be approximated by a [[secant]]. Given this geometrical interpretation, it is not surprising that derivatives can be used to determine many geometrical properties of graphs of functions, such as [[concavity]] or [[convexity]].

==GMA Network programs==

It should be noted that not all functions have derivatives. For example, functions do not have derivatives at points where they have either a vertical tangent or a [[continuity (mathematics)|discontinuity]]. However, functions may fail to have derivatives even if they are continuous and have no vertical tangents.

===Local programs===

====[[GMA News and Public Affairs]]====

[[Image:GMANewsLogo.png|right|200px]]

*''[[100% Pinoy]]

*''[[24 Oras]]

*''[[Emergency (TV program)|Emergency]]

*''[[GMA Flash Report]]

*''[[GMA Weekend Report]]

*''[[Imbestigador]]

*''[[I-Witness]]

*''[[Kapuso Mo, Jessica Soho]]

*''[[Kay Susan Tayo]]

*''[[Palaban]]

*''[[Pinoy Meets World]]

*''[[Reporter's Notebook]]

*''[[Saksi]]

*''[[Sine Totoo]]

*''[[Unang Hirit]]

====GMA KiliTV====

==Differentiation and differentiability==

[[Image:gma kilitv.jpg|right|150px]]

*''[[Bahay Mo Ba 'To]]

*''[[Bitoy's Funniest Videos]]

*''[[Bubble Gang]]

*''[[Daddy Di Do Du]]

*''[[HP: Ibang Level Na!]]

*''[[Nuts Entertainment]]

*''[[Who's Your Daddy Now?]]

====GMA Telebabad====

In somewhat dated language, '''differentiation''' expresses the rate at which one quantity ''y'' changes as a result of a change in another quantity ''x'' on which it has a [[function (mathematics)|functional relationship]]. Using the symbol Δ to refer to change in a quantity, this rate is defined as a [[limit]] of difference quotients

*''[[Impostora]]

*''[[Lupin (Philippine TV series)|Lupin]]

*''[[Mga Mata ni Anghelita]]

====GMA Dramarama sa Hapon====

:<math> \frac{\Delta y}{\Delta x} </math>

*''[[Daisy Siete|Daisy Siete Season 15: Isla Chikita]]

*''[[Pati Ba Pintig ng Puso]]

*''[[Sinasamba Kita]]

====Sabado Star Power====

as &Delta; ''x'' approaches 0. In [[Leibniz|Leibniz]]'s notation, the derivative of ''y'' with respect to ''x'' is written

*''[[Fantastic Man]]

*''Kapuso Sine Special

:<math> \frac{dy}{dx} </math>

*''[[Startalk]]

*''[[Wish Ko Lang]]

*''[[Walang Tulugan with the Master Showman]]

====Linggo Bingo====

suggesting the ratio of two [[infinitesimal]] quantities. The above expression is pronounced in various ways such as "''dy over dx''". The form "''dy dx''" is also used conversationally, although it may be confused with the notation for element of area.

*''[[All Star K!]]

*''[[Boys Nxt Door]]

*''[[Mel and Joey]]

*''[[Mga Kuwento ni Lola Basyang]]

*''[[Move: The Search For Billy Crawford's Pinoy Dancers]]

*''[[Showbiz Central]]

*''SNBO (Sunday Night Box Office)

*''[[SOP Rules]]

*''[[Tok! Tok! Tok! Isang Milyon Pasok!]]

====Other programs====

In contemporary mathematical language, one dispenses with referring to dependent quantities and simply states that differentiation is a mathematical operation on functions. The precise definition of this operation (which also dispenses with referring to ''infinitesimal'' quantities) is given as the limit as ''h'' approaches 0 of

*''[[Art Angel]]

*''[[Eat Bulaga!]]

*''[[Diyos at Bayan]]

*''[[Kapwa Ko Mahal Ko]]

*''[[Lovely Day: LD Force]]

*''[[Magpakailanman]]

*''[[Maynila (TV series)|Maynila]]

*''[[Sis (Philippine TV program)|Sis]]

*''[[Just Joking]]

===Foreign/canned shows===

:<math> \frac{f(x+h) - f(x)}{h}. </math>

====Asianovelas====

*''[[Great Teacher Onizuka#Live-Action|GTO: Live-Action]]''

*''[[Jumong (TV series)|Jumong]]

*''[[Love Truly]]''

*''[[Love in Heaven]]''

====Animé====

This definition is discussed in more detail below. If ''f'' is a function, the derivative of the function ''f'' at the value ''x'' is written in several ways:

*''[[Bleach (manga)|Bleach]]''

*''[[Card Captor Sakura]]''

*''[[Case Closed|Detective Conan]]''

*''[[Chrono Crusade]]''

*''[[Dragon Ball Z]]''

*''[[Fullmetal Alchemist]]''

*''[[Ghostfighter]]''

*''[[Fighting Spirit|Knockout]]''

*''[[Machine Robo Rescue]]''

*''[[One Piece]]

*''[[Pokemon]]''

*''[[Rockman Exe]]''

*''[[Samurai 7]]

*''[[The Slayers]]''

*''[[Slamdunk]]''

====Tokusatsu====

*<math> f'(x) \quad</math>

*''[[Kamen Rider Blade]]''

pronounced "''f [[prime (mark)|prime]] of x''"

*''[[Genseishin Justiriser]]''

*<math> \frac{d}{dx} f (x) </math>

pronounced "''d by d x of f of x''" or "''d d x of f of x''".

*<math> \frac{df}{dx} </math>

pronounced "''d f by d x''" or "''d f d x''"

*<math> D_x f \quad </math>

pronounced "''d sub x of f''".

===RGMA programs nationwide===

A function is '''differentiable''' at a point ''x'' if its derivative exists at that point; a function is differentiable on an [[Interval (mathematics)|interval]] if it is differentiable at every ''x'' within the interval. If a function is not [[Continuous function|continuous]] at ''c'', then there is no slope and the function is therefore not differentiable at ''c''; however, even if a function is continuous at ''c'', it may not be differentiable.

*Newscasts

**''[[Balitang Bisdak]]'' -regional newscast of GMA Cebu

**''[[Ratsada]]'' - regional newscast of GMA Iloilo and Bacolod

**''[[Testigo]]'' -regional newscast of GMA Davao

*Variety

**''[[Bongga!]]'' - regional variety show produced by GMA Entertainment Group for GMA Iloilo and Bacolod

**''[[Kuyaw!]]''- regional variety show produced by GMA Entertainment Group for GMA Davao

**''[[Oi! (TV program)|Oi!]]''- regional variety show produced by GMA Entertainment Group for GMA Cebu

==GMA Network upcoming programs==

The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a '''second derivative'''. Similarly, the derivative of a second derivative is a '''third derivative''', and so on.

===Local Programs===

*Angela Markado<ref name="multiple"/>

*[[Celebrity Duets (Philippine TV series)|Celebrity Duets]]<ref>{{cite web | last = Barrameda | first = Joe | title = John Estrada, idolo ng kalalakihan | publisher = Abante Tonight | date =[[June 12]], [[2007]] | url =http://abante-tonite.com/issue/june1107/showbiz_jb.htm | accessdate = June 12 | language = [[Filipino language|Filipino]]}}</ref> (July 2007)

*Dyesebel<ref name="multiple"/>

*[[Fulhaus]]<ref>{{cite web | last = Diones | first = Allan | title = FIRST READ ON PEP: Pia Guanio is Vic Sotto's leading lady in new GMA-7 sitcom "Fulhaus" | publisher = Philippine Entertainment Portal | date =[[June 28]], [[2007]] | url =http://www.pep.ph/news/13828/FIRST-READ-ON-PEP:-Pia-Guanio-is-Vic-Sottos-leading-lady-in-new-GMA-7-sitcom-Fulhaus | accessdate = June 28}}</ref> (September 2007)

*Igorota<ref name="multiple"/>

*[[Kamandag]]<ref name="multiple"/> (2008)

*[[L.U.V. Pow]]<ref>{{cite web | last = Erece | first = Dinno | title = GMA-7 unleashes its new shows for 2007 | publisher = Philippine Entertainment Portal | date =[[January 19]], [[2007]] | url =http://www.pep.ph/articles/news/11932.htm | accessdate = January 27 }}</ref> (September 2007)

*[[Magic Kamison|Magic Kamison: Season 2]]<ref> (2008)[http://pep.ph/news/12505/Alex-de-Rossi-ready-to-work-with-Jennylyn-Mercado-in-&%2339;Super-Twins& Alex de Rossi ready to work with Jennylyn Mercado in 'Super Twins'], ''pep.ph'', n.d. Accessed last May 21, 2007.</ref> (2008)

*[[Marimar (Philippine TV series)|Marimar]]<ref name="multiple">{{cite web | last = Lorenzo | first = Alfie | title = GMA 7, siksik sa mga pasabog | publisher = Abante Tonight | date =[[February 5]], [[2007]] | url =http://www.abante-tonite.com/issue/feb0507/showbiz_al.htm | accessdate = February 22 | language = [[Filipino language|Filipino]]}}</ref> (September 2007)

*Rosang Agimat<ref name="multiple"/>

*[[Shaider (Philippine TV series)|Shaider]]<ref>[http://www.pep.ph/news/13214/GMA-7-to-serialize-Mga-Mata-ni-Angelita-of-the-late-Julie-Vega "GMA-7 to serialize Mga Mata ni Angelita of the late Julie Vega"], '' mb.com.ph'', n.d. Accessed last April 12, 2007.</ref> (2007)

*[[Sine Novela]]<ref name="multiple"/>

# [[Kung Mahawi Man Ang Ulap]]

# Saan Darating Ang Umaga

# Dapat Ka Bang Mahalin

# Gaano Kadalas Ang Minsan

# Saan Nagtatago Ang Pag-ibig

# Paano Ba Ang Mangarap

# Kung Aagawin Mo Ang Lahat Sa Akin

# Tayong Dalawa

# Ikaw Naman Ang Iiyak

*[[StarStruck|StarStruck: The Star Explosion]]<ref>{{cite web | last = E. Bautista | first = Mario | title = Starstruck Top 6 to get maximum exposure | publisher = Journal Online | date =[[April 10]], [[2007]] | url =http://www.journal.com.ph/index.php?issue=2007-04-10&sec=3&aid=14564 | accessdate = June 6}}</ref> (December 2007)

*[[Super Twins#Season 2|Super Twins: Season 2]]<ref>{{cite web | last = Asiñas | first = Noel | title = Na-‘X’ na ‘MKM’ episode, lumusot sa MTRCB! | publisher = Journal Online | date =[[June 6]], [[2007]] | url =http://www.journal.com.ph/index.php?issue=2007-06-06&sec=3&aid=20470 | accessdate = June 6}}</ref> (March 2008)

*[[Tasya Fantasya]]<ref name="multiple"/>

*Untitled Philippine-China Co-Production<ref>{{cite web | last = Asiñas | first = Noel | title = China makikisosyo sa GMA-7 | publisher = Journal Online | date =[[June 13]], [[2007]] | url =http://www.journal.com.ph/index.php?issue=2007-06-14&sec=3&aid=21319 | accessdate = June 13}}</ref>

*Zimatar<ref name="multiple"/>

==Special Shows==

==Newton's difference quotient==

*[[Kapusolympics]]: The GMA 57th Anniversary Special (July 1, 2007)

==References==

[[Image:Secant-calculus.png|thumb|300px| Secant to curve ''y''= ''f''(''x'') determined by points (''x'', ''f''(''x'')) and (''x''+''h'', ''f''(''x''+''h'')).]]

<!--<nowiki>

<!-- [[Image:Derivative.png|right]] -->

See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref> and </ref> tags, and the template below.

Derivatives are defined by taking the [[limit (mathematics)|limit]] of the slope of [[secant]] lines as they approach a [[tangent]]

</nowiki>-->

line. Simply put, the derivative of a function will show the slope of the tangent line to any given point x, thus allowing it to be used to calculate other definitions of f(x).

{{reflist}}

==See also==

It is hard to directly find the slope of the [[tangent]] line to a given function because we only know one point on it, the [[Image:Tangent-calculus.png|thumb|300px|Tangent line at (''x'', ''f''(''x''))]] point where it is tangent to the function. Instead we will approximate the tangent line by [[secant]] lines. When we take the [[limit (mathematics)|limit]] of the slopes of the nearby secant lines, we will get the slope of the tangent line.

*[[GMA Network]]

To find the slopes of the nearby secant lines, choose a small number ''h''. ''h'' represents a small change in ''x'', and it can be either positive or negative. The slope of the line through the points (''x'',''f(x)'') and (''x+h'',''f(x+h)'') is

*[[List of Philippine television shows]]

:<math>{f(x+h)-f(x)\over h}.</math>

*[[GMA Pinoy TV]]

This expression is [[Isaac Newton|Newton]]'s '''difference quotient'''. The '''derivative of''' '''''f''''' '''at''' '''''x''''' is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:

*[[List of programs broadcast by Q]]

:<math>f'(x)=\lim_{h\to 0}{f(x+h)-f(x)\over h}.</math>

*[[List of shows previously aired by GMA Network]]

==External links==

[[Image:Lim-secant.png|thumb|300px|Tangent line as limit of secants.]]

*[http://www.igma.tv Official Website of GMA Network]

If the derivative of ''f'' exists at every point ''x'', we can define the '''derivative of''' '''''f''''' to be the function whose value at a point ''x'' is the derivative of ''f'' at ''x''.

*[http://www.gmapinoytv.com Official Website of GMA Pinoy TV]

{{Programs}}

Since immediately [[substitution|substituting]] 0 for ''&#8710;x'' results in [[division by zero]], calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the ''h'' in the [[denominator]] can be cancelled. This happens very easily for [[polynomial]]s; see [[calculus with polynomials]]. For almost all functions, however, the result is a mess. Fortunately there are general rules which make it easy to differentiate most functions that are easy to write down; see below.

[[Category:GMA Network| ]]

See [[Derivative (examples)]] for some examples of how to use this quotient.

[[Category:Lists of television series by network|GMA Network]]

==Notations for differentiation==

The simplest notation for differentiation that is in current use is due to [[Joseph_Louis_Lagrange|Lagrange]] and uses the [[Prime (mark)|prime]], &prime;. To take derivatives of ''f''(''x'') at the point ''a'', we write:

:''f'' &prime;(''a'') for the first derivative,

:''f'' &Prime;(''a'') for the second derivative,

:''f'' &Prime;&prime;(''a'') for the third derivative and then

:''f''^(''n'')(''a'') for the ''n''th derivative (''n'' > 3).

For the function whose value at each ''x'' is the derivative of ''f''(''x''), we write ''f'' &prime;(''x''). Similarly, for the second derivative of ''f'' we write ''f'' &Prime;(''x''), and so on.

The other common notation for differentiation is due to [[Leibniz]]. For the function whose value at ''x'' is the derivative of ''f'' at ''x'', we write:

:<math>\frac{d\left(f(x)\right)}{dx}.</math>

We can write the derivative of ''f'' at the point ''a'' in two different ways:

<!-- You have no idea how hard it was to make Wikipedia display this. Why can't I use \Bigg? This is such an ugly hack. -->

:<math>\frac{d\left(f(x)\right)}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \left(\frac{d\left(f(x)\right)}{dx}\right)(a).</math>

If the output of ''f''(''x'') is another variable, for example, if ''y''=''f''(''x''), we can write the derivative as:

:<math>\frac{dy}{dx}.</math>

Higher derivatives are expressed as

:<math>\frac{d^n\left(f(x)\right)}{dx^n}</math> or <math>\frac{d^ny}{dx^n}</math>

for the ''n''-th derivative of ''f''(''x'') or ''y'' respectively. Historically, this came from the fact that, for example, the 3rd derivative is:

:<math>\frac{d \left(\frac{d \left( \frac{d \left(f(x)\right)} {dx}\right)} {dx}\right)} {dx}</math>

which we can loosely write as:

:<math>\left(\frac{d}{dx}\right)^3 \left(f(x)\right) =

\frac{d^3}{\left(dx\right)^3} \left(f(x)\right).</math>

Dropping brackets gives the notation above.

Leibniz's notation is versatile in that it allows one to specify the variable for differentiation (in the denominator). This is especially relevant for [[partial derivative|partial differentiation]]. It also makes the [[chain rule]] easy to remember, because the "d" terms appear symbolically to cancel:

:<math>\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>

However, it is important to remember that the "d" terms ''cannot'' literally cancel, because on their own they are undefined. They are only defined when used together to express a derivative.

[[Newton]]'s notation for differentiation was to place a dot over the function name:

:<math>\dot{x} = \frac{dx}{dt} = x'(t)</math>

:<math>\ddot{x} = x''(t)</math>

and so on.

Newton's notation is mainly used in [[mechanics]], normally for time derivatives such as velocity and acceleration, and in [[ODE]] theory. It is usually only used for first and second derivatives.

==Critical points==

Points on the [[graph of a function|graph]] of a function where the derivative is undefined or equals [[zero]] are called ''critical points'' or sometimes ''[[stationary point]]s'' (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a [[local minimum]]; if negative, it is a [[local maximum]]; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in [[Optimization (mathematics)|optimization]]. In fact, local minima and maxima can only occur at critical points. This is related to the [[extreme value theorem]].

==Notable derivatives==

* For [[Natural_logarithm|logarithmic functions]]:

** The derivative of ''[[E (mathematical constant)|e]]''^''x'' is ''e''^''x''

** The derivative of [[natural logarithm|ln]] ''x'' is 1/''x''.

* For [[trigonometric function]]s

** The derivative of sin ''x'' is cos ''x''.

** The derivative of cos ''x'' is -sin ''x''.

** The derivative of tan ''x'' is sec² ''x''.

** The derivative of csc ''x'' is -(csc ''x'')(cot ''x'').

** The derivative of sec ''x'' is (sec ''x'')(tan ''x'').

** The derivative of cot ''x'' is -csc² ''x''.

==Physics==

Arguably the most important application of calculus to [[physics]] is the concept of the "'''time derivative'''"&nbsp;&mdash; the rate of change over time&nbsp;&mdash; which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

* [[Velocity]] (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.

* [[Acceleration]] is the derivative (with respect to time) of an object's velocity.

* [[Jerk]] is the derivative (with respect to time) of an object's acceleration.

For example, if an object's position <math>p(t) = -16t^2 + 16t + 32</math>; then, the object's velocity is <math>\dot p(t) = p'(t) = -32t + 16</math>; the object's acceleration is <math>\ddot p(t) = p''(t) = -32</math>; and the object's jerk is <math>p'''(t) = 0.</math>

If the [[velocity]] of a [[automobile|car]] is given, as a function of [[time]]; then, the derivative of said function with respect to time describes the [[acceleration]] of said car, as a function of time.

==Algebraic manipulation==

Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via [[algebra]]ic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of ''proving'' the following "powerful differentiation rules"; these rules are derived from the difference quotient.

* ''Constant rule'': The derivative of any [[constant]] is [[zero]].

** ''Constant multiple rule'': If ''c'' is some [[real number]]; then, the derivative of <math>cf(x)</math> equals ''c'' [[multiplication|multiplied]] by the derivative of ''f''(''x'') (a consequence of linearity below)

* ''[[Linearity of differentiation|Linearity]]'': (af + bg)' = af ' + bg' for all functions ''f'' and ''g'' and all real numbers ''a'' and ''b''.

* ''[[Calculus with polynomials|General power rule]]'' (''Polynomial rule''): If <math>f(x) = x^r</math>, for some [[real number]] ''r''; <math>f'(x) = rx^{r-1}.</math>

* ''[[Product rule]]'': (fg)' = f 'g + fg' for all functions ''f'' and ''g''.

* ''[[Quotient rule]]'': (f/g)' = (f 'g - fg')/<math>(g^2)</math> unless g is zero.

* ''[[Chain rule]]'': If f(x) = h(g(x)), then f '(x) = h'[g(x)] * g'(x).

* ''[[Inverse functions and differentiation]]'': If <math>y = f(x)</math>, <math>x = f^{-1}(y)</math>, and f(''x'') and its inverse are differentiable, with <math>dy/dx</math> non-zero, then <math>dx/dy = 1/(dy/dx).</math>

* ''Derivative of one variable with respect to another when both are functions of a third variable'': Let <math>x = f(t)</math> and <math>y = g(t)</math>. Now <math>d y/d x = (d y/d t)/(d x/d t).</math>

* ''[[Implicit differentiation]]'': If <math>f(x,y) = 0</math> is an implicit function, we have: d''y''/d''x'' = - (&part;f / &part;''x'') / (&part;f / &part;''y'').

In addition, the derivatives of some common functions are useful to know. See the [[table of derivatives]].

As an example, the derivative of

:<math>f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7</math>

is

:<math>f'(x) = 8x^3 + 2x\cos (x^2) - (1/x)\;e^x - \ln (x)\;e^x.</math>

== Using derivatives to graph functions ==

Derivatives are a useful tool for examining the [[graph of a function|graphs of functions]]. In particular, the points in the interior of the ___domain of a real-valued function which take that function to local [[extremum|extrema]] will all have a first derivative of zero. However, not all critical points are local extrema; for example, ''f(x)=x³'' has a critical point at ''x=0'', but it has neither a maximum nor a minimum there. The [[first derivative test]] and the [[second derivative test]] provide ways to determine if the critical points are maxima, minima or neither.

In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the [[eigenvalue]]s of the [[Hessian matrix]] of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3).

Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional ___domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is [[continuity (mathematics)|continuous]]) will have values in between its values at the critical points on either side.

== Generalizations ==

Where a function depends on more than one variable, the concept of a '''[[partial derivative]]''' is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as &part;/&part;x (where &part; is a rounded 'd' known as the 'partial derivative symbol'). Some people say the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.

The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable [[manifold]]s; the derivative at a certain point then becomes a [[linear transformation]] between the corresponding [[tangent space]]s and the derivative function becomes a map between the [[tangent bundle]]s.

In order to differentiate all [[continuous]] functions and much more, one defines the concept of [[distribution]].

For [[Complex number|complex]] functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function ''f''(''x''&nbsp;+&nbsp;''iy'')&nbsp;= ''x''&nbsp;+&nbsp;2''iy'' satisfies the latter, but not the first. See also [[Holomorphic function]].

==See also==

*[[table of derivatives]]

*[[differintegral]]

*[[covariant derivative]]

*[[Lie derivative]]

*[[exterior derivative]]

*[[exterior covariant derivative]]

*[[derivation (abstract algebra)]]

*[[functional derivative]]

*[[Kähler differential]]

*[[Pincherle derivative]]

*[[Fréchet derivative]]

*[[partial derivative]]

*[[q-derivative]]

*[[directional derivative]]

*[[total derivative]]

*[[convective derivative]]

*[[inner derivative]]

*[[Tangent_space#The_derivative_of_a_map|pushforward]]

*[[Schwarzian derivative]]

==External links==

* [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives.

==References==

* [[Ron Larson|Larson, Ron]], [[Robert P. Hostetler|Hostetler, Robert P.]] and [[Bruce H. Edwards|Edwards, Bruce H.]] ([[2003]]). ''[[Calculus of a Single Variable: Early Transcendental Functions]]'' (3rd Edition). Houghton Mifflin Company. ISBN 061822307X [http://college.hmco.com/mathematics/larson/calculus_early/3e/instructors/ ''textbook site online'']

* Anton, Howard (1980). ''Calculus with analytical geometry.''. New York:John Wiley and Sons. ISBN 0-471-03248-4

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