Definition:Derivative/Real Function/Derivative on Interval

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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 with respect to time:

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

is sometimes used.