Category:Definitions/Higher Derivatives

This category contains definitions related to Higher Derivatives.
Related results can be found in Category: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 $\map {f} \xi$ 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 $f^{\paren {n - 1} }$.