Definition:Derivative/Higher Derivatives/Third Derivative/Notation

Definition

The third derivative of $\map f x$ is variously denoted as:

$\map {f'''} x$

$\map {f^{\paren 3} } x$

$D^3 \map f x$

$D_{xxx} \map f x$

$\dfrac {\d^3} {\d x^3} \map f x$

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

$y''' := \map {\dfrac \d {\d x} } {\dfrac {\d^2 y} {\d x^2} }$

and written:

$\dfrac {\d^3 y} {\d x^3}$

Leibniz's notation for the third derivative of a function $y = \map f x$ with respect to the independent variable $x$ is:

$\dfrac {\d^3 y} {\d x^3}$

Newton's notation for the third derivative of a function $y = \map f t$ with respect to the independent variable $t$ is:

$\map {\dddot f} t$

or:

$\dddot y$

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