Definition:Derivative/Higher Derivatives/Third Derivative

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.

Also denoted as
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:
 * $\dfrac {\mathrm d^3 y}{\mathrm d x^3}$

If the derivative is with respect to time:


 * $\dddot f \left({x}\right)$
 * $\dddot y$

is sometimes used.

Also see

 * Definition:Differentiability Class
 * Definition:Order of Derivative