# Definition:Derivative

## Definition

Informally, a **derivative** is the rate of change of one variable with respect to another.

### Real Function

Let $I$ be an open real interval.

Let $f: I \to \R$ be a real function defined on $I$.

Let $\xi \in I$ be a point in $I$.

Let $f$ be differentiable at the point $\xi$.

#### Definition 1

That is, suppose the limit $\ds \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$ exists.

Then this limit is called the **derivative of $f$ at the point $\xi$**.

#### Definition 2

That is, suppose the limit $\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists.

Then this limit is called the **derivative of $f$ at the point $\xi$**.

### Complex Function

Let $D\subseteq \C$ be an open set.

Let $f : D \to \C$ be a complex function.

Let $z_0 \in D$ be a point in $D$.

Let $f$ be complex-differentiable at the point $z_0$.

That is, suppose the limit $\ds \lim_{h \mathop \to 0} \ \frac {\map f {z_0 + h} - \map f {z_0} } h$ exists.

Then this limit is called the **derivative of $f$ at the point $z_0$**.

### Vector-Valued Function

Let $U \subset \R$ be an open set.

Let $\map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k: U \to \R^n$ be a vector-valued function.

Let $\mathbf f$ be differentiable at $u \in U$.

That is, let each $f_j$ be differentiable at $u \in U$.

The **derivative of $\mathbf f$ with respect to $x$ at $u$** is defined as

- $\map {\dfrac {\d \mathbf f} {\d x} } u = \ds \sum_{k \mathop = 1}^n \map {\dfrac {\d f_k} {\d x} } u \mathbf e_k$

where $\map {\dfrac {\d f_k} {\d x} } u$ is the derivative of $f_k$ with respect to $x$ at $u$.

### Function With Values in Normed Space

Let $U \subset \R$ be an open set.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.

Let $f : U \to X$ be differentiable at $x \in U$.

The **derivative of $f$ at $x$** is defined as the element $\map {f'} x \in X$ which satisfies:

- $\ds \lim_{h \mathop \to 0} \norm {\frac {\map f {x + h} - \map f x} h - \map {f'} x}_X = 0$

## Notation

There are various notations available to be used for the derivative of a function $f$ with respect to the independent variable $x$:

- $\dfrac {\d f} {\d x}$

- $\map {\dfrac \d {\d x} } f$

- $\dfrac {\d y} {\d x}$ when $y = \map f x$

- $\map {f'} x$

- $\map {D f} x$

- $\map {D_x f} x$

When evaluated at the point $\tuple {x_0, y_0}$, the derivative of $f$ at the point $x_0$ can be variously denoted:

- $\map {f'} {x_0}$

- $\map {D f} {x_0}$

- $\map {D_x f} {x_0}$

- $\map {\dfrac {\d f} {\d x} } {x_0}$

- $\valueat {\dfrac {\d f} {\d x} } {x \mathop = x_0}$

and so on.

### Leibniz Notation

Leibniz's notation for the derivative of a function $y = \map f x$ with respect to the independent variable $x$ is:

- $\dfrac {\d y} {\d x}$

### Newton Notation

Newton's notation for the derivative of a function $y = \map f t$ with respect to the independent variable $t$ is:

- $\map {\dot f} t$

or:

- $\dot y$

which many consider to be less convenient than the Leibniz notation.

This notation is usually reserved for the case where the independent variable is time.

## Higher Derivatives

### Second Derivative

Let $f$ be a real function which is differentiable on an open interval $I$.

Hence $f'$ is defined on $I$ as the derivative of $f$.

Let $\xi \in I$ be a point in $I$.

Let $f'$ be differentiable at the point $\xi$.

Then the **second derivative** $\map {f''} \xi$ is defined as:

- $\ds f'' := \lim_{x \mathop \to \xi} \dfrac {\map {f'} x - \map {f'} \xi} {x - \xi}$

### Third Derivative

Let $f$ be a real function which is twice differentiable on an open interval $I$.

Let $f''$ denote the second derivate.

Then the **third derivative** $f'''$ is defined as:

- $f''' := \dfrac {\d} {\d x} f'' = \map {\dfrac {\d} {\d x} } {\dfrac {\d^2} {\d x^2} f}$

### Higher Order Derivatives

Higher order derivatives are defined in similar ways:

The $n$th derivative of a function $y = \map f x$ is defined as:

- $\map {f^{\paren n} } x = \dfrac {\d^n y} {\d x^n} := \begin {cases} \map {\dfrac \d {\d x} } {\dfrac {\d^{n - 1} y} {\d x^{n - 1} } } & : n > 0 \\ y & : n = 0 \end {cases}$

assuming appropriate differentiability for a given $f^{\paren {n - 1} }$.

### First Derivative

If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the **first derivative**:

Let $I \subset \R$ be an open interval.

Let $f: I \to \R$ be a real function.

Let $f$ be differentiable on the interval $I$.

Then the **derivative of $f$** is the real function $f': I \to \R$ whose value at each point $x \in I$ is the derivative $\map {f'} x$:

- $\ds \forall x \in I: \map {f'} x := \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

## Order of Derivative

The **order** of a derivative is the **number of times it has been differentiated**.

For example:

- a first derivative is of
**first order**, or**order $1$** - a second derivative is of
**second order**, or**order $2$**

and so on.

## Ordinary Derivative

Such a derivative as has been described here is known as an **ordinary derivative**.

This is to distinguish it from a Partial Derivative, which applies to functions of more than one independent variables.

## Also known as

Some sources refer to a **derivative** as a **differential coefficient**, and abbreviate it **D.C.**

Some sources call it a **derived function**.

## Also see

- Results about
**derivatives, and differential calculus in general**can be found here.

## Historical Note

The rigorous treatment of a derivative was developed by Carl Friedrich Gauss, Niels Henrik Abelâ€Ž and Augustin Louis Cauchy.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1956: E.L. Ince:
*Integration of Ordinary Differential Equations*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions - 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**derived function**