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 $\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ 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 $f' \left({x}\right)$:

$\displaystyle \forall x \in I: f' \left({x}\right) := \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$


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:

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

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:

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


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.


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 $y = f \left({x}\right)$ with respect to the independent variable $x$ is:

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


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