Category:Higher Derivatives
This category contains results about Higher Derivatives.
Definitions specific to this category can be found in Definitions/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 $f^{\paren {n - 1} }$.
Subcategories
This category has the following 5 subcategories, out of 5 total.
E
O
- Order of Derivative (empty)
S
- Second Derivatives (empty)
T
- Third Derivatives (empty)
Z
- Zeroth Derivatives (empty)