# 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^{}

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

$${f}^{\prime \prime}(x)=\underset{h\to 0}{lim}\frac{{f}^{\prime}(x+h)-{f}^{\prime}(x)}{h}\mathit{\hspace{1em}}\mathrm{for}\mathrm{all}x\in I.$$ |

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

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

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

If also ${f}^{\prime \prime}$ is a differentiable function, its derivative function is denoted by ${f}^{\prime \prime \prime}$ and called the 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 , 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 .

Title | higher order derivatives |

Canonical name | HigherOrderDerivatives |

Date of creation | 2013-03-22 16:46:30 |

Last modified on | 2013-03-22 16:46:30 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 26B05 |

Classification | msc 26A24 |

Related topic | HigherOrderDerivativesOfSineAndCosine |

Related topic | TaylorSeriesOfHyperbolicFunctions |

Defines | derivative function |

Defines | first derivative |

Defines | second derivative |

Defines | order of derivative |

Defines | differentiation |

Defines | differentiate |

Defines | twice differentiable |