higher order derivatives

Let the real function f be defined and differentiableMathworldPlanetmathPlanetmath on the open intervalDlmfPlanetmath I.  Then for every  xI,  there exists the value f(x) as a certain real number.  This means that we have a new functionMathworldPlanetmath

xf(x), (1)

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


or simply f.

Forming the derivative function of a function is called differentiation, the corresponding verb is 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 twice differentiable.  Formally,


The function  xf′′(x)  is called the or the second derivative of f.  Similarly, one can call (1) the of f.

Example.  The first derivative of  xx3  is  x3x2  and the second derivative is  x6x,  since


If also f′′ is a differentiable function, its derivative function is denoted by f′′′ and called the of f, and so on.

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