# Definition:Derivative/Higher Derivatives

## 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:

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

### 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 $f$.

## Sources

- 1961: David V. Widder:
*Advanced Calculus*(2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.2$ Derivatives