Definition:Derivative/Real Function/Derivative on Interval

Definition
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 $\map {f'} x$:
 * $\ds \forall x \in I: \map {f'} x := \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

Also denoted as
It can be variously denoted as:
 * $\dfrac {\d f} {\d x}$


 * $\map {\dfrac \d {\d x} } f$


 * $\map {f'} x$


 * $D \map f x$


 * $D_x \map f x$

If the derivative is time:


 * $\map {\dot f} x$
 * $\dot f$

is sometimes used.