Definition:Derivative/Higher Derivatives/Second Derivative/Notation
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Definition
The second derivative is variously denoted as:
- $\map {f''} \xi$
- $D^2 \map f \xi$
- $D_{xx} \map f \xi$
- $\map {\dfrac {\d^2} {\d x^2} } \xi$
If $y = \map f x$, then it can also expressed as $y''$:
- $y'' := \map {\dfrac \d {\d x} } {\dfrac {\d y} {\d x} }$
and written:
- $\dfrac {\d^2 y} {\d x^2}$
Leibniz Notation
Leibniz's notation for the second derivative of a function $y = \map f x$ with respect to the independent variable $x$ is:
- $\dfrac {\d^2 y} {\d x^2}$
Newton Notation
Newton's notation for the second derivative of a function $y = \map f t$ with respect to the independent variable $t$ is:
- $\map {\ddot f} t$
or:
- $\ddot y$
This notation is usually reserved for the case where the independent variable is time.
Also see
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: $13.43$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.2$
- 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): Notation and terminology