# Definition:Derivative/Higher Derivatives/Second Derivative

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## Definition

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

Hence $f'$ is defined on $I$ as the derivative of $f$.

Let $\xi \in I$ be a point in $I$.

Let $f'$ be differentiable at the point $\xi$.

Then the **second derivative** $\map {f' '} \xi$ is defined as:

- $\ds f' ' := \lim_{x \mathop \to \xi} \dfrac {\map {f'} x - \map {f'} \xi} {x - \xi}$

Thus the **second derivative** is defined as the derivative of the first derivative.

If $f'$ is differentiable, then it is said that $f$ is **doubly differentiable**, or **twice differentiable**.

## Also defined as

The **second derivative** may also be seen defined as:

- $\ds \map {f' '} \xi = \lim_{h \mathop \to 0} \frac {\map {f'} {\xi + h} - \map {f'} \xi} h$

## Notation

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:

- $\dfrac {\d^2 y} {\d x^2}$

## Also see

- Results about
**second 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: $13.43$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 10.2$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**order**:**1.**(of a derivative) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**order**:**1.**(of a derivative) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**second derivative**