Definition:Derivative/Higher Derivatives/Second Derivative/Notation

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Definition

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:

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


Leibniz Notation

Leibniz's notation for the second derivative of a function $y = f \left({x}\right)$ with respect to the independent variable $x$ is:

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


Newton Notation

Newton's notation for the second derivative of a function $y = f \left({t}\right)$ with respect to the independent variable $t$ is:

$\ddot f \left({t}\right)$

or:

$\ddot y$


This notation is usually reserved for the case where the independent variable is time.


Also see


Sources