# Definition:Derivative/Higher Derivatives/Second Derivative

## 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** $f'' \left({\xi}\right)$ is defined as:

- $\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {f' \left({x}\right) - f' \left({\xi}\right)} {x - \xi}$

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

It may also be seen defined as:

- $\displaystyle f'' \left({\xi}\right) = \lim_{h \mathop \to 0} \frac {f' \left({\xi + h}\right) - f' \left({\xi}\right)} h$

## Notation

The **second derivative** is variously denoted as:

- $f'' \left({\xi}\right)$

- $D^2 f \left({\xi}\right)$

- $D_{xx} f \left({\xi}\right)$

- $\dfrac{\mathrm d^2}{\mathrm d x^2} \left({\xi}\right)$

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

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

and written:

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

## 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): $\S 10.2$