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 $\displaystyle \lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ exists.

Then this limit is called the derivative of $f$ at the point $\xi$ and 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 $y = f \left({x}\right)$ it can be written as:


 * $\left.{\dfrac {\mathrm dy} {\mathrm dx}}\right \vert_{x = \xi}$

Alternatively it may be written:
 * $\displaystyle f' \left({\xi}\right) = \lim_{h \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$

The two definitions are equivalent.