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单词 HigherOrderDerivatives
释义

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

f:I

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,

f′′(x)=limh0f(x+h)-f(x)hforallxI.

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

ddx(3x2)=23x2-1=6x.

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 .

Titlehigher order derivatives
Canonical nameHigherOrderDerivatives
Date of creation2013-03-22 16:46:30
Last modified on2013-03-22 16:46:30
OwnerPrimeFan (13766)
Last modified byPrimeFan (13766)
Numerical id7
AuthorPrimeFan (13766)
Entry typeDefinition
Classificationmsc 26B05
Classificationmsc 26A24
Related topicHigherOrderDerivativesOfSineAndCosine
Related topicTaylorSeriesOfHyperbolicFunctions
Definesderivative function
Definesfirst derivative
Definessecond derivative
Definesorder of derivative
Definesdifferentiation
Definesdifferentiate
Definestwice differentiable
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更新时间:2025/5/4 2:58:11