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 $f' \left({x}\right)$:
 * $\displaystyle \forall x \in I: f' \left({x}\right) := \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$

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


 * $\dfrac {\mathrm d} {\mathrm d x} \left({f}\right)$


 * $f' \left({x}\right)$


 * $D f \left({x}\right)$


 * $D_x f \left({x}\right)$

If the derivative is with respect to time:


 * $\dot f \left({x}\right)$
 * $\dot f$

is sometimes used.