# 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 $\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists.

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

## 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.