# Difference between revisions of "Derivative"

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− | One of the basic concepts in mathematical analysis. Suppose that a real-valued function | + | 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 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. | |

− | + | 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 derivative]]s, 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 $\phi=f'\circ h$ <ref name="Choquet" />. | |

− | |||

− | |||

+ | ====References==== | ||

− | + | <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> | |

− | |||

− | |||

− | < |

## Revision as of 11:45, 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

**How to Cite This Entry:**

Derivative.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Derivative&oldid=39010