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 $\xi$.
Definition 1
That is, suppose the limit $\ds \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$ exists.
Then this limit is called the derivative of $f$ at the point $\xi$.
Definition 2
That is, suppose the limit $\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists.
Then this limit is called the derivative of $f$ at the point $\xi$.
Notation
There are various notations available to be used for the derivative of a function $f$ with respect to the independent variable $x$:
- $\dfrac {\d f} {\d x}$
- $\map {\dfrac \d {\d x} } f$
- $\dfrac {\d y} {\d x}$ when $y = \map f x$
- $\map {f'} x$
- $\map {D f} x$
- $\map {D_x f} x$
When evaluated at the point $\tuple {x_0, y_0}$, the derivative of $f$ at the point $x_0$ can be variously denoted:
- $\map {f'} {x_0}$
- $\map {D f} {x_0}$
- $\map {D_x f} {x_0}$
- $\map {\dfrac {\d f} {\d x} } {x_0}$
- $\valueat {\dfrac {\d f} {\d x} } {x \mathop = x_0}$
and so on.
Leibniz Notation
Leibniz's notation for the derivative of a function $y = \map f x$ with respect to the independent variable $x$ is:
- $\dfrac {\d y} {\d x}$
Newton Notation
Newton's notation for the derivative of a function $y = \map f t$ with respect to the independent variable $t$ is:
- $\map {\dot f} t$
or:
- $\dot y$
which many consider to be less convenient than the Leibniz notation.
This notation is usually reserved for the case where the independent variable is time.