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:
- $f' := \dfrac {\d} {\d x} f = \map {\dfrac {\d} {\d x} } {\dfrac {\d^2} {\d x^2} f}$
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.
Notation
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.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Higher Derivatives: $13.44$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.2$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): third derivative