Definition:Derivative/Higher Derivatives/Second Derivative

From ProofWiki
Jump to: navigation, search

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 $f'' \left({\xi}\right)$ is defined as:

$\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {f' \left({x}\right) - f' \left({\xi}\right)} {x - \xi}$


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

It may also be seen defined as:

$\displaystyle f'' \left({\xi}\right) = \lim_{h \mathop \to 0} \frac {f' \left({\xi + h}\right) - f' \left({\xi}\right)} h$


Notation

The second derivative is variously denoted as:

$f'' \left({\xi}\right)$
$D^2 f \left({\xi}\right)$
$D_{xx} f \left({\xi}\right)$
$\dfrac{\mathrm d^2}{\mathrm d x^2} \left({\xi}\right)$


If $y = f \left({x}\right)$, then it can also expressed as $y''$:

$y'' := \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac {\mathrm d y} {\mathrm d x}}\right)$

and written:

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


Also see


Sources