\begin{definition}[Definition:Derivative/Real Function/Derivative on Interval]
Let $I \subset \R$ be an open interval.
Let $f: I \to \R$ be a real function.
Let $f$ be differentiable on the interval $I$.
Then the '''derivative of $f$''' is the real function  $f': I \to \R$ whose value at each point $x \in I$ is the derivative $\map {f'} x$:
:$\ds \forall x \in I: \map {f'} x := \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$
\end{definition}