Definition:Derivative/Higher Derivatives/Third Derivative

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Definition

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:

$f''' := \dfrac {\mathrm d} {\mathrm d x} f'' = \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac{\mathrm d^2}{\mathrm d x^2} f}\right)$


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.


Notation

The third derivative of $f \left({x}\right)$ is variously denoted as:

$f''' \left({x}\right)$
$D^3 f \left({x}\right)$
$D_{xxx} f \left({x}\right)$
$\dfrac{\mathrm d^3}{\mathrm d x^3} f \left({x}\right)$


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

$y''' := \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac {\mathrm d^2 y} {\mathrm d x^2}}\right)$

and written:

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


Also see


Sources