# Definition:Derivative/Real Function/Derivative at Point

## 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 $\displaystyle \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$ exists.

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

## Also denoted as

The derivative of $f$ at the point $\xi$ is variously denoted:

$f' \left({\xi}\right)$
$D f \left({\xi}\right)$
$D_x f \left({\xi}\right)$
$\dfrac {\mathrm d} {\mathrm d x} \left({\xi}\right)$

If the derivative is with respect to time:

$\dot f \left({\xi}\right)$

is sometimes used.