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单词 PartialDerivative
释义

partial derivative


The partial derivativeMathworldPlanetmath of a multivariable function f is simply its derivative with respect to only one variable, keeping all other variables constant (which are not functions of the variable in question). The formal definition is

Dif(𝐚)=fai=limh01h(f(a1ai+han)-f(𝐚))=limh0f(𝐚+hei)-f(𝐚)h

where ei is the standard basis vector of the ith variable. Since this only affects the ith variable, one can derive the function using common rules and tables, treating all other variables (which are not functions of ai) as constants. For example, if f(𝐱)=x2+2xy+y2+y3z, then

(1)fx=2x+2y(2)fy=2x+2y+3y2z(3)fz=y3

Note that in equation (1), we treated y as a constant, since we were differentiating with respect to x. (d(c*x)dx=c) The partial derivative of a vector-valued functionPlanetmathPlanetmath f(𝐱) with respect to variable ai is a vector Di𝐟=fai.
Multiple Partials:
Multiple partial derivatives can be treated just like multiple derivatives. There is an additional degree of freedom though, as you can compound derivatives with respect to different variables. For example, using the above function,

(4)2fx2=x(2x+2y)=2(5)2fzy=z(2x+2y+3y2z)=3y2(6)2fyz=y(y3)=3y2

D12 is another way of writing x1x2. If f(𝐱) is continuousMathworldPlanetmath in the neighborhoodMathworldPlanetmathPlanetmath of 𝐱,and Dijf and Djif are continuous in an open set V, it can be shown(see Clairaut’s theorem (http://planetmath.org/ClairautsTheorem)) that Dijf(𝐱)=Djif(𝐱) in V, where i,j are the ith and jth variables. In fact, as long as an equal number of partials are taken with respect to each variable, changing the order of differentiationMathworldPlanetmath will produce the same results in the above condition.
Another form of notation is f(a,b,c,)(𝐱) where a is the partial derivative with respect to the first variable a times, b is the partial with respect to the second variable b times, etc.

Titlepartial derivative
Canonical namePartialDerivative
Date of creation2013-03-22 11:58:30
Last modified on2013-03-22 11:58:30
OwnerMathprof (13753)
Last modified byMathprof (13753)
Numerical id26
AuthorMathprof (13753)
Entry typeDefinition
Classificationmsc 26B12
Related topicDerivative2
Related topicDerivativeNotation
Related topicJacobianMatrix
Related topicDirectionalDerivative
Related topicGradient
Related topicHessianMatrix
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更新时间:2025/5/3 12:13:52