higher order derivatives HigherOrderDerivatives 2007-03-02 11:16:29 2009-08-19 14:59:15 Definition derivative derivative function first derivative second derivative order of derivative differentiation differentiate twice differentiable % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} \PMlinkescapeword{order} Let the real function $f$ be defined and differentiable on the open interval $I$.\, Then for every\, $x \in I$,\, there exists the value $f'(x)$ as a certain real number.\, This means that we have a new function \begin{align} x \mapsto f'(x), \end{align} the so-called {\em derivative function} of $f$; it is denoted by $$f':\, I\to \mathbb{R}$$ or simply $f'$. Forming the derivative function of a function is called {\em differentiation}, the corresponding verb is {\em differentiate}. If the derivative function $f'$ is differentiable on $I$, then we have again a new function, the derivative function of the derivative function of $f$, which is denoted by $f''$.\, Then $f$ is said to be \emph{twice differentiable}.\, Formally, $$f''(x) = \lim_{h\to 0}\frac{f'(x+h)-f'(x)}{h}\quad \mathrm{for\,all\,}\,x\in I.$$ The function\, $x\mapsto f''(x)$\, is called the \PMlinkescapetext{{\em second order derivative}} or {\em the second derivative} of $f$.\, Similarly, one can call (1) the \PMlinkescapetext{{\em first (order) derivative}} of $f$. \textbf{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 $f''$ is a differentiable function, its derivative function is denoted by $f'''$ and called the \PMlinkescapetext{{\em third (order) derivative}} of $f$, and so on. Generally, $f$ can have the derivatives of first, second, third, \ldots, $n$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 \PMlinkescapetext{addition}, it's sometimes convenient to think that the $0${\em 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 \PMlinkescapetext{orders}.