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 $\map {f''} \xi$ is defined as:
 * $\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {\map {f'} x - \map {f'} \xi} {x - \xi}$

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
It may also be seen defined as:
 * $\displaystyle \map {f''} \xi = \lim_{h \mathop \to 0} \frac {\map {f'} {\xi + h} - \map {f'} \xi} h$

Also see

 * Definition:Differentiability Class
 * Definition:Order of Derivative