Definition:Derivative/Higher Derivatives

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Definition

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 $\map {f} \xi$ is defined as:

$\ds f := \lim_{x \mathop \to \xi} \dfrac {\map {f'} x - \map {f'} \xi} {x - \xi}$


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:

$f' := \dfrac {\d} {\d x} f = \map {\dfrac {\d} {\d x} } {\dfrac {\d^2} {\d x^2} f}$


Higher Order Derivatives

Higher order derivatives are defined in similar ways:


The $n$th derivative of a function $y = \map f x$ is defined as:

$\map {f^{\paren n} } x = \dfrac {\d^n y} {\d x^n} := \begin {cases}

\map {\dfrac \d {\d x} } {\dfrac {\d^{n - 1} y} {\d x^{n - 1} } } & : n > 0 \\ y & : n = 0 \end {cases}$

assuming appropriate differentiability for a given $f^{\paren {n - 1} }$.


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 $\map {f'} x$:

$\ds \forall x \in I: \map {f'} x := \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$


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.


Zeroth Derivative

The zeroth derivative of a real function $f$ is defined as $f$ itself:

$f^{\paren 0} := f$

where $f^{\paren n}$ denotes the $n$th derivative of $f$.


Examples

Example: $x^4$

Consider the equation:

$\forall y \in \R: y = x^4$

Then:

\(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds 4 x^3\) Derivative of Power
\(\ds \dfrac {\d^2 y} {\d x^2}\) \(=\) \(\ds 12 x^2\) Derivative of Power
\(\ds \dfrac {\d^3 y} {\d x^3}\) \(=\) \(\ds 24 x\) Derivative of Power


Also see

  • Results about higher derivatives can be found here.


Sources

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