Definition:Derivative/Real Function

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Definition

At a Point

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 $\xi$.


On an Open Interval

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 $\map {f'} x$:

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


With Respect To

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

Let $f$ be defined as an equation: $y = \map f x$.


Then the derivative of $y$ with respect to $x$ is defined as:

$\ds y^\prime = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = D_x \, \map f x$

This is frequently abbreviated as derivative of $y$ WRT or w.r.t. $x$, and often pronounced something like wurt.


We introduce the quantity $\delta y = \map f {x + \delta x} - \map f x$.

This is often referred to as the small change in $y$ consequent on the small change in $x$.


Hence the motivation behind the popular and commonly-seen notation:

$\ds \dfrac {\d y} {\d x} := \lim_{\delta x \mathop \to 0} \dfrac {\map f {x + \delta x} - \map f x} {\delta x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x}$


Hence the notation $\map {f^\prime} x = \dfrac {\d y} {\d x}$.

This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes.

However, it has the disadvantage that the variable $x$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done.

For practical applications, however, this is not usually a problem.


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.

Such a derivative is also known as an ordinary derivative.

This is to distinguish it from a partial derivative, which applies to functions of more than one independent variable.


In his initial investigations into differential calculus, Isaac Newton coined the term fluxion to mean derivative.


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.


Sources