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# higher order derivatives

Let the real function $f$ be defined and differentiable on the open interval $I$. Then for every $x\in I$, there exists the value $f^{{\prime}}(x)$ as a certain real number. This means that we have a new function

$\displaystyle x\mapsto f^{{\prime}}(x),$ | (1) |

the so-called derivative function of $f$; it is denoted by

$f^{{\prime}}:\,I\to\mathbb{R}$ |

or simply $f^{{\prime}}$.

Forming the derivative function of a function is called differentiation, the corresponding verb is differentiate.

If the derivative function $f^{{\prime}}$ is differentiable on $I$, then we have again a new function, the derivative function of the derivative function of $f$, which is denoted by $f^{{\prime\prime}}$. Then $f$ is said to be *twice differentiable*. Formally,

$f^{{\prime\prime}}(x)=\lim_{{h\to 0}}\frac{f^{{\prime}}(x+h)-f^{{\prime}}(x)}{% h}\quad\mathrm{for\,all\,}\,x\in I.$ |

The function $x\mapsto f^{{\prime\prime}}(x)$ is called the second order derivative or the second derivative of $f$. Similarly, one can call (1) the first (order) derivative of $f$.

Example. The first derivative of $x\mapsto x^{3}$ is $x\mapsto 3x^{2}$ and the second derivative is $x\mapsto 6x$, since

$\frac{d}{dx}(3x^{2})=2\cdot 3x^{{2-1}}=6x.$ |

If also $f^{{\prime\prime}}$ is a differentiable function, its derivative function is denoted by $f^{{\prime\prime\prime}}$ and called the third (order) derivative of $f$, and so on.

Generally, $f$ can have the derivatives of first, second, third, …, $n$th order, where $n$ may be an arbitrarily big positive integer. If $n$ is four or greater, the $n$th derivative of $f$ is usually denoted by $f^{{(n)}}$. In addition, it’s sometimes convenient to think that the $0$th order derivative $f^{{(0)}}$ of $f$ is the function $f$ itself.

The phrase “$f$ is infinitely differentiable” means that $f$ has the derivatives of all orders.

## Mathematics Subject Classification

26B05*no label found*26A24

*no label found*

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