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

$$f^{\prime }:I\to ℝ$$

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  $x\mapsto 3x^2$  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 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