Definition:Derivative

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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 Abel‎ and Augustin Louis Cauchy.


Sources

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