Definition:Derivative/Higher Derivatives/Second Derivative

From ProofWiki
Jump to navigation Jump to search


This page has been identified as a candidate for refactoring of basic complexity.
In particular: "Also defined as" could come in as a separate definition. Don't forget the {{SourceReview}}
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


Definition

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:

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


Thus the second derivative is defined as the derivative of the first derivative.


If $f'$ is differentiable, then it is said that $f$ is doubly differentiable, or twice differentiable.


Also defined as

The second derivative may also be seen defined as:

$\ds \map {f' '} \xi = \lim_{h \mathop \to 0} \frac {\map {f'} {\xi + h} - \map {f'} \xi} h$


Notation

The second derivative is variously denoted as:

$\map {f' '} \xi$
$D^2 \map f \xi$
$D_{xx} \map f \xi$
$\map {\dfrac {\d^2} {\d x^2} } \xi$


If $y = \map f x$, then it can also expressed as $y' '$:

$y' ' := \map {\dfrac \d {\d x} } {\dfrac {\d y} {\d x} }$

and written:

$\dfrac {\d^2 y} {\d x^2}$


Also see

  • Results about second derivatives can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Derivative/Higher_Derivatives/Second_Derivative&oldid=706750"