Definition:Derivative/Higher Derivatives/Second Derivative

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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:

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


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

It may also be seen defined as:

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


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


Sources