Definition:Derivative/Higher Derivatives

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

Hence $f^\prime$ is defined as above.

Let $f^\prime$ be differentiable on the interval $I$.

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

Let $f^\prime$ be differentiable at the point $\xi$.

Then the second derivative $f^{\prime \prime} \left({\xi}\right)$ is defined as:
 * $\displaystyle \lim_{x \to \xi} \frac {f^\prime \left({x}\right) - f^\prime \left({\xi}\right)} {x - \xi}$

Again, it is variously denoted:
 * $f^{\prime \prime} \left({\xi}\right)$


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


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


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

And again, it may alternatively be written:
 * $\displaystyle f^{\prime \prime} \left({\xi}\right) = \lim_{h \to 0} \frac {f^\prime \left({\xi + h}\right) - f^{\prime} \left({\xi}\right)} h$

Thus the second derivative is defined as the derivative of the derivative (which, in this context, can be referred to as the first derivative).

If $y = f \left({x}\right)$, then it can also denoted by $y''$ or $\dfrac{\mathrm d^2y}{\mathrm d x^2}$.

If $f^\prime$ is differentiable, then it is said that $f$ is doubly differentiable, or twice differentiable.

Higher Order Derivatives
Higher order derivatives are defined in similar ways.

In general, the notation for the $n$th derivative at a point $\xi$ is given by:
 * $f^{\left({n}\right)} \left({\xi}\right)$


 * $D^n f \left({\xi}\right)$


 * $D_{x \left({n}\right)} f \left({\xi}\right)$


 * $\dfrac{\mathrm d^n}{\mathrm d x^n} \left({\xi}\right)$

The $n$ in $f^{\left({n}\right)}$ is sometimes written as a roman numeral, but this is considered on this website as being laughably archaic and ridiculous.

If the $n$th derivative exists for a function, then $f$ is described as being $n$ times differentiable.

See differentiability class.

First Derivative
If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the first derivative.

Zeroth Derivative
Some sources define the zeroth derivative of a real function as the function itself:


 * $f^{\left({0}\right)} := f$