Definition:Derivative/Real Function/Derivative at Point

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Definition

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 $\xi$.


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.


Also see