# Derivative

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.

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 $\phi=f'\circ h$ .