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

Di⁢f⁢(𝐚)=∂⁡f∂⁡ai=limh→0⁡1h⁢(f⁢(a1⋮ai+h⋮an)-f⁢(𝐚))=limh→0⁡f⁢(𝐚+h⁢e→i)-f⁢(𝐚)h

where e→i 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+2⁢x⁢y+y2+y3⁢z, then

(1)∂⁡f∂⁡x=2⁢x+2⁢y(2)∂⁡f∂⁡y=2⁢x+2⁢y+3⁢y2⁢z(3)∂⁡f∂⁡z=y3

Note that in equation (1), we treated y as a constant, since we were differentiating with respect to x. (d⁢(c*x)d⁢x=c) The partial derivative of a vector-valued functionPlanetmathPlanetmath f→⁢(𝐱) with respect to variable ai is a vector Di⁢𝐟→=∂⁡f→∂⁡ai.
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)∂2⁡f∂⁡x2=∂∂⁡x⁢(2⁢x+2⁢y)=2(5)∂2⁡f∂⁡z⁢∂⁡y=∂∂⁡z⁢(2⁢x+2⁢y+3⁢y2⁢z)=3⁢y2(6)∂2⁡f∂⁡y⁢∂⁡z=∂∂⁡y⁢(y3)=3⁢y2

D12 is another way of writing ∂∂⁡x1⁢∂⁡x2. If f⁢(𝐱) is continuousMathworldPlanetmath in the neighborhoodMathworldPlanetmathPlanetmath of 𝐱, and Di⁢j⁢f and Dj⁢i⁢f are continuous in an open set V, it can be shown (see Clairaut’s theorem (http://planetmath.org/ClairautsTheorem)) that Di⁢j⁢f⁢(𝐱)=Dj⁢i⁢f⁢(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.

Title partial derivative
Canonical name PartialDerivative
Date of creation 2013-03-22 11:58:30
Last modified on 2013-03-22 11:58:30
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 26
Author Mathprof (13753)
Entry type Definition
Classification msc 26B12
Related topic Derivative2
Related topic DerivativeNotation
Related topic JacobianMatrix
Related topic DirectionalDerivative
Related topic Gradient
Related topic HessianMatrix