Definition:Derivative
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Definition
Real Functions
Derivative at a Point
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 $\xi$.
That is, suppose the limit $\displaystyle \lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ exists.
Then this limit is called the derivative of $f$ at the point $\xi$ and is variously denoted:
- $f^\prime \left({\xi}\right)$
- $D f \left({\xi}\right)$
- $D_x f \left({\xi}\right)$
- $\dfrac {\mathrm d} {\mathrm d x} \left({\xi}\right)$
If $y = f \left({x}\right)$ it can be written as:
- $\left.{\dfrac {\mathrm dy} {\mathrm dx}}\right \vert_{x=\xi}$
Alternatively it may be written:
- $\displaystyle f^\prime \left({\xi}\right) = \lim_{h \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$
Derivative on an Interval
Let $f$ be a real function defined on an open interval $I$.
Let $f$ be differentiable on the interval $I$.
Then $f^\prime: I \to \R$ is defined as the real function whose value at each point $x \in I$ is $f^\prime \left({x}\right)$.
It can be variously denoted as:
- $\dfrac {\mathrm d f}{\mathrm d x}$
- $\dfrac {\mathrm d} {\mathrm d x} \left({f}\right)$
- $f^\prime \left({x}\right)$
- $D f \left({x}\right)$
- $D_x f \left({x}\right)$
With Respect To
Let $f$ be a real function which is differentiable on an open interval $I$.
Let $f$ be defined as an equation: $y = f \left({x}\right)$.
Then the derivative of $y$ with respect to $x$ is defined as:
- $\displaystyle y^\prime = \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h = D_x f \left({x}\right)$
This is frequently abbreviated as derivative of $y$ WRT or w.r.t. $x$, and often pronounced something like wurt.
We introduce the quantity $\delta y = f \left({x + \delta x}\right) - f \left({x}\right)$.
This is often referred to as the small change in $y$ consequent on the small change in $x$.
Hence the motivation behind the popular and commonly-seen notation:
- $\displaystyle \frac{\mathrm d y}{\mathrm d x} := \lim_{\delta x \to 0} \frac {f \left({x + \delta x}\right) - f \left({x}\right)} {\delta x} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x}$
Hence the notation $f^\prime \left({x}\right) = \dfrac{\mathrm d y}{\mathrm d x}$. This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes.
However, it has the disadvantage that the variable $x$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done. For practical applications, however, this is not usually a problem.
Complex Functions
The definition for a complex function is similar to that for real functions.
Let $f \left({z}\right): \C \to \C$ be a single-valued continuous complex function in a domain $D \subseteq \C$.
Let $z_0 \in D$ be a point in $D$.
Let $f$ be complex-differentiable at the point $z_0$.
That is, suppose the limit $\displaystyle \lim_{h \to 0} \ \frac {f \left({z_0 + h}\right) - f \left({z_0}\right)} h$ exists as a finite number and is independent of how the complex increment $h$ tends to $0$.
Then this limit is called the derivative of $f$ at the point $z_0$ and is variously denoted:
- $f^\prime \left({z_0}\right)$
- $D f \left({z_0}\right)$
- $D_z f \left({z_0}\right)$
- $\dfrac{\mathrm d f}{\mathrm d z} \left({z_0}\right)$
Further, let $f$ be complex-differentiable at all points in $D$.
Then $f^\prime: D \to \C$ is defined as the complex function whose value at each point $z \in D$ is $f^\prime \left({z}\right)$.
It can be variously denoted as:
- $\dfrac {\mathrm d f}{\mathrm d z}$
- $\dfrac {\mathrm d}{\mathrm d z} \left({f}\right)$
- $f^\prime \left({z}\right)$
- $D f \left({z}\right)$
- $D_z f \left({z}\right)$
Higher Derivatives
Second Derivative
Let $f$ be a real function which is differentiable on an open interval $I$.
Hence $f^\prime$ is defined as above.
Let $f^\prime$ be differentiable on the interval $I$.
Let $\xi \in I$ be a point in $I$.
Let $f^\prime$ be differentiable at the point $\xi$.
Then the second derivative $f^{\prime \prime} \left({\xi}\right)$ is defined as:
- $\displaystyle \lim_{x \to \xi} \frac {f^\prime \left({x}\right) - f^\prime \left({\xi}\right)} {x - \xi}$
Again, it is variously denoted:
- $f^{\prime \prime} \left({\xi}\right)$
- $D^2 f \left({\xi}\right)$
- $D_{xx} f \left({\xi}\right)$
- $\dfrac{\mathrm d^2}{\mathrm d x^2} \left({\xi}\right)$
And again, it may alternatively be written:
- $\displaystyle f^{\prime \prime} \left({\xi}\right) = \lim_{h \to 0} \frac {f^\prime \left({\xi + h}\right) - f^{\prime} \left({\xi}\right)} h$
Thus the second derivative is defined as the derivative of the derivative (which, in this context, can be referred to as the first derivative).
If $y = f \left({x}\right)$, then it can also denoted by $y''$ or $\dfrac{\mathrm d^2y}{\mathrm d x^2}$.
If $f^\prime$ is differentiable, then it is said that $f$ is doubly differentiable, or twice differentiable.
Higher Order Derivatives
Higher order derivatives are defined in similar ways.
In general, the notation for the $n$th derivative at a point $\xi$ is given by:
- $f^{\left({n}\right)} \left({\xi}\right)$
- $D^n f \left({\xi}\right)$
- $D_{x \left({n}\right)} f \left({\xi}\right)$
- $\dfrac{\mathrm d^n}{\mathrm d x^n} \left({\xi}\right)$
The $n$ in $f^{\left({n}\right)}$ is sometimes written as a roman numeral, but this is considered on this website as being laughably archaic and ridiculous.
If the $n$th derivative exists for a function, then $f$ is described as being $n$ times differentiable.
First Derivative
If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the first derivative.
Order of a Derivative
The order of a derivative is the number of times it has been differentiated.
For example:
- a first derivative is of first order, or order $1$
- a second derivative is of second order, or order $2$
and so on.
Ordinary Derivative
Such a derivative as has been described here is known as a ordinary derivative.
This is to distinguish it from a Partial Derivative, which applies to functions of more than one independent variables.
Also see
