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 $\xi$.

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 $\xi$

Also defined as
The derivative of $f$ at the point $\xi$ may also be written:
 * $\displaystyle f' \left({\xi}\right) := \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$

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.

Also see

 * Equivalence of Definitions of Derivative.