Definition:Derivative/Higher Derivatives/Third Derivative

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Let $f$ be a real function which is twice differentiable on an open interval $I$.

Let $f' '$ denote the second derivate.

Then the third derivative $f' ' '$ is defined as:

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

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

If $f' '$ is differentiable, then it is said that $f$ is triply differentiable, or thrice differentiable.


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:

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

Also see

  • Results about third derivatives can be found here.