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 $$\underset{x\to x_0}{lim}\frac{f(x)-f(x_0)}{x-x_0}$$ This limit is called the derivative of the function $f$ at the point $x_0$. If one sets $y=f(x)$, $$x-x_0=\mathrm{\Delta }x,f(x)-f(x_0)=f(x_0+\mathrm{\Delta }x)-f(x_0)=\mathrm{\Delta }y$$ then the limit $\text{(???)}$ can be written as: $$\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}.$$

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 $$\underset{\begin{array}{c}x⟶x_0,\\ x\in E\end{array}}{lim}\frac{f(x)-f(x_0)}{x-x_0}.$$

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[edit]

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].

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