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One of the basic concepts in mathematical analysis. Suppose that a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312601.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312602.png" /> is defined in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312603.png" /> and that there exists a finite or infinite limit
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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One of the basic concepts in mathematical analysis. Suppose that a real-valued function $f$ of a real variable $x$ is defined in a neighborhood of a point $x_0$ and that there exists a finite or infinite limit
 +
\begin{equation}
 +
\label{eq:1}
 +
\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}
 +
\end{equation}
 +
This limit is called the derivative of the function $f$ at the point $x_0$. If one sets $y=f(x)$,
 +
\begin{equation}
 +
x-x_0=\Delta x,\quad f(x)-f(x_0)=f(x_0+\Delta x)-f(x_0)=\Delta y
 +
\end{equation}
 +
then the limit \eqref{eq:1} can be written as:
 +
\begin{equation}
 +
\lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x}.
 +
\end{equation}
  
This limit is called the derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312605.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312606.png" />. If one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312607.png" />,
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Also the notations $f'(x_0)$, $\frac{df(x_0)}{dx}$, $\frac{dy}{dx}$, $(\frac{d}{dx})f(x_0)$, and some others are used to denote this limit.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312608.png" /></td> </tr></table>
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The operation of computing the derivative is called [[differentiation]]. If the derivative $f'(x_0)$ is finite, $f$ is said to be differentiable at the point $x_0$. A function differentiable at every point of a set is said to be differentiable on that set. A differentiable function is always continuous. However, there are continuous functions that have no derivative at any point of a given interval (see [[Non-differentiable function]]).
  
then the limit (*) can be written as:
+
Let a function $f$ be differentiable in an [[interval]]. Its derivative $f'$ may turn out to be a [[discontinuous function]]. However, according to [[Baire classes|Baire's classification]] it is always a function of the first class and has the [[Darboux property]]: If it takes two values, it takes every intermediate value as well.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d0312609.png" /></td> </tr></table>
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A generalization of the concept of the derivative is the concept of a derivative over a set. Suppose that a real-valued function $f$ is defined on a set $E$ of real numbers, that $x_0$ is a limit point of $E$, that $x_0\in E$, and that there exists a finite or infinite limit
 +
\begin{equation}
 +
\lim_{\substack{x\longrightarrow x_0, \\ x\in E}} \frac{f(x)-f(x_0)}{x-x_0}.
 +
\end{equation}
  
Also the notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126013.png" />, and some others are used to denote this limit.
+
This limit is called the derivative of $f$ over the set $E$ at the point $x_0$ and is denoted by the symbol $f'_{E}(x_0)$. The derivative of a function over a set is a generalization of the concept of a derivative. Variations of the generalization are the concept of a [[one-sided derivative]], a [[Dini derivative]], and an [[approximate derivative]].
  
The operation of computing the derivative is called differentiation. If the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126014.png" /> is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126015.png" /> is said to be differentiable at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126016.png" />. A function differentiable at every point of a set is said to be differentiable on that set. A differentiable function is always continuous. However, there are continuous functions that have no derivative at any point of a given interval (see [[Non-differentiable function|Non-differentiable function]]).
+
The above definition of the derivative (and its generalizations), as well as simple properties of it, extend almost without change to complex-valued and vector-valued functions of a real or complex variable. Moreover, there exists a concept of a derivative of a scalar-valued point function in an [[Euclidean space]] $\mathbb{R}^{n}$ (see [[Gradient]]), and of a derivative of a set function with respect to a measure (in particular, with respect to area, volume, etc.). The concept of a derivative is extended to vector-valued point functions in an abstract space (see [[Differentiation of a mapping]]).
  
Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126017.png" /> be differentiable in an interval. Its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126018.png" /> may turn out to be a [[Discontinuous function|discontinuous function]]. However, according to Baire's classification (see [[Baire classes|Baire classes]]) it is always a function of the first class and has the Darboux property: If it takes two values, it takes every intermediate value as well.
+
For a geometric and mechanical interpretation of the derivative, the simplest rules of differentiation, higher derivatives, [[partial derivative]]s, and also for references see [[Differential calculus]].
  
A generalization of the concept of the derivative is the concept of a derivative over a set. Suppose that a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126019.png" /> is defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126020.png" /> of real numbers, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126021.png" /> is a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126022.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126023.png" />, and that there exists a finite or infinite limit
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126024.png" /></td> </tr></table>
 
  
This limit is called the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126025.png" /> over the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126026.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126027.png" /> and is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126028.png" />. The derivative of a function over a set is a generalization of the concept of a derivative. Variations of the generalization are the concept of a [[One-sided derivative|one-sided derivative]], a [[Dini derivative|Dini derivative]], and an [[Approximate derivative|approximate derivative]].
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====Comments====
  
The above definition of the derivative (and its generalizations), as well as simple properties of it, extend almost without change to complex-valued and vector-valued functions of a real or complex variable. Moreover, there exists a concept of a derivative of a scalar-valued point function in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126029.png" /> (see [[Gradient|Gradient]]), and of a derivative of a set function with respect to a measure (in particular, with respect to area, volume, etc.). The concept of a derivative is extended to vector-valued point functions in an abstract space (see [[Differentiation of a mapping|Differentiation of a mapping]]).
+
G. Choquet has proved that a function $\phi$ on $[a,b]$ is of the first Baire class and has the Darboux property (if and) only if there exists a differentiable function $f$ on $[a,b]$ and a [[homeomorphism]] $h$ of $[a,b]$ such that $\phi=f'\circ h$ <ref name="Choquet" />.
  
For a geometric and mechanical interpretation of the derivative, the simplest rules of differentiation, higher derivatives, partial derivatives, and also for references see [[Differential calculus|Differential calculus]].
 
  
  
 +
====References====
  
====Comments====
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<references>
G. Choquet has proved that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126031.png" /> is of the first Baire class and has the Darboux property (if and) only if there exists a differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126033.png" /> and a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031260/d03126036.png" />. See [[#References|[a1]]] for details and references.
+
<ref name="Choquet">G. Choquet,  "Outils topologiques et métriques de l'analyse mathématique" , Centre Docum. Univ. Paris  (1969)  (Rédigé par C. Mayer) {{MR|0262426}} {{ZBL|}}</ref>
 
+
</references>
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Choquet,  "Outils topologiques et métriques de l'analyse mathématique" , Centre Docum. Univ. Paris  (1969)  (Rédigé par C. Mayer)</TD></TR></table>
 

Latest revision as of 12:47, 5 July 2016


One of the basic concepts in mathematical analysis. Suppose that a real-valued function $f$ of a real variable $x$ is defined in a neighborhood of a point $x_0$ and that there exists a finite or infinite limit \begin{equation} \label{eq:1} \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0} \end{equation} This limit is called the derivative of the function $f$ at the point $x_0$. If one sets $y=f(x)$, \begin{equation} x-x_0=\Delta x,\quad f(x)-f(x_0)=f(x_0+\Delta x)-f(x_0)=\Delta y \end{equation} then the limit \eqref{eq:1} can be written as: \begin{equation} \lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x}. \end{equation}

Also the notations $f'(x_0)$, $\frac{df(x_0)}{dx}$, $\frac{dy}{dx}$, $(\frac{d}{dx})f(x_0)$, and some others are used to denote this limit.

The operation of computing the derivative is called differentiation. If the derivative $f'(x_0)$ is finite, $f$ is said to be differentiable at the point $x_0$. A function differentiable at every point of a set is said to be differentiable on that set. A differentiable function is always continuous. However, there are continuous functions that have no derivative at any point of a given interval (see Non-differentiable function).

Let a function $f$ be differentiable in an interval. Its derivative $f'$ may turn out to be a discontinuous function. However, according to Baire's classification it is always a function of the first class and has the Darboux property: If it takes two values, it takes every intermediate value as well.

A generalization of the concept of the derivative is the concept of a derivative over a set. Suppose that a real-valued function $f$ is defined on a set $E$ of real numbers, that $x_0$ is a limit point of $E$, that $x_0\in E$, and that there exists a finite or infinite limit \begin{equation} \lim_{\substack{x\longrightarrow x_0, \\ x\in E}} \frac{f(x)-f(x_0)}{x-x_0}. \end{equation}

This limit is called the derivative of $f$ over the set $E$ at the point $x_0$ and is denoted by the symbol $f'_{E}(x_0)$. The derivative of a function over a set is a generalization of the concept of a derivative. Variations of the generalization are the concept of a one-sided derivative, a Dini derivative, and an approximate derivative.

The above definition of the derivative (and its generalizations), as well as simple properties of it, extend almost without change to complex-valued and vector-valued functions of a real or complex variable. Moreover, there exists a concept of a derivative of a scalar-valued point function in an Euclidean space $\mathbb{R}^{n}$ (see Gradient), and of a derivative of a set function with respect to a measure (in particular, with respect to area, volume, etc.). The concept of a derivative is extended to vector-valued point functions in an abstract space (see Differentiation of a mapping).

For a geometric and mechanical interpretation of the derivative, the simplest rules of differentiation, higher derivatives, partial derivatives, and also for references see Differential calculus.


Comments

G. Choquet has proved that a function $\phi$ on $[a,b]$ is of the first Baire class and has the Darboux property (if and) only if there exists a differentiable function $f$ on $[a,b]$ and a homeomorphism $h$ of $[a,b]$ such that $\phi=f'\circ h$ [1].


References

  1. G. Choquet, "Outils topologiques et métriques de l'analyse mathématique" , Centre Docum. Univ. Paris (1969) (Rédigé par C. Mayer) MR0262426
How to Cite This Entry:
Derivative. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Derivative&oldid=13836
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article