Definition:Derivative

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Definition

Real Function

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 $\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ exists.


Then this limit is called the derivative of $f$ at the point $\xi$.


Definition 2

That is, suppose the limit $\displaystyle \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$ exists.


Then this limit is called the derivative of $f$ at the point $\xi$.


Complex Function

Let $D\subseteq \C$ be an open set.

Let $f : D \to \C$ be a complex function.

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.


Then this limit is called the derivative of $f$ at the point $z_0$.


Vector-Valued Function

Let $U\subset\R$ be an open set.

Let $f = (f_1, \ldots, f_n)^\intercal : U \to \R^n$ be a vector-valued function.

Let $x\in U$.

Let $f$ be differentiable at $x$.

That is, let each $f_j$ be differentiable at $x$.


The derivative of $f$ at $x$ is $(f_1'(x), \ldots, f_n'(x))^\intercal$, where $f_j'(x)$ is the derivative of $f_j$ at $x$.


Notation

There are various notations available to be used for the derivative of a function $f$ with respect to the independent variable $x$:

  • $\dfrac {\mathrm d f} {\mathrm d x}$
  • $\dfrac {\mathrm d} {\mathrm d x} \left({f}\right)$
  • $\dfrac {\mathrm d y} {\mathrm d x}$ when $y = f \left({x}\right)$
  • $f' \left({x}\right)$
  • $D f \left({x}\right)$
  • $D_x f \left({x}\right)$


When evaluated at the point $\left({x_0, y_0}\right)$, the derivative of $f$ at the point $x_0$ can be variously denoted:

  • $f' \left({x_0}\right)$
  • $D f \left({x_0}\right)$
  • $D_x f \left({x_0}\right)$
  • $\dfrac {\mathrm d f} {\mathrm d x} \left({x_0}\right)$

and so on.


Higher Derivatives

Second Derivative

Let $f$ be a real function which is differentiable on an open interval $I$.

Hence $f'$ is defined on $I$ as the derivative of $f$.


Let $\xi \in I$ be a point in $I$.

Let $f'$ be differentiable at the point $\xi$.

Then the second derivative $f'' \left({\xi}\right)$ is defined as:

$\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {f' \left({x}\right) - f' \left({\xi}\right)} {x - \xi}$


Third Derivative

Let $f$ be a real function which is twice differentiable on an open interval $I$.

Let $f''$ denote the second derivate.


Then the third derivative $f'''$ is defined as:

$f''' := \dfrac {\mathrm d} {\mathrm d x} f'' = \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac{\mathrm d^2}{\mathrm d x^2} f}\right)$


Higher Order Derivatives

Higher order derivatives are defined in similar ways:


The $n$th derivative of a function $y = f \left({x}\right)$ is defined as:

$f^{\left({n}\right)} \left({x}\right) = \dfrac {\mathrm d^n y} {\mathrm d x^n} := \begin{cases} \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac {\mathrm d^{n-1}y} {\mathrm d x^{n-1} } }\right) & : n > 0 \\ y & : n = 0 \end{cases}$

assuming appropriate differentiability for a given $f^{\left({n-1}\right)}$.


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:


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

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


Order of 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 an ordinary derivative.

This is to distinguish it from a Partial Derivative, which applies to functions of more than one independent variables.


Also see


Historical Note

The rigorous treatment of a derivative was developed by Carl Friedrich Gauss, Niels Henrik Abel‎ and Augustin Louis Cauchy.


Sources