chain rule (several variables)

The chain ruleMathworldPlanetmath is a theorem of analysis that governs derivatives of composed functions. The basic theorem is the chain rule for functions of one variables (see here ( This entry is devoted to the more general version involving functions of several variables and partial derivativesMathworldPlanetmath. Note: the symbol Dk will be used to denote the partial derivative with respect to the kth variable.

Let F(x1,,xn) and G1(x1,,xm),,Gn(x1,,xm) be differentiable functions of several variables, and let


be the function determined by the compositionMathworldPlanetmath of F with G1,,Gn The partial derivatives of H are given by


The chain rule can be more compactly (albeit less precisely) expressed in terms of the Jacobi-Legendre partial derivative symbols ( note). Just as in the Leibniz system, the basic idea is that of one quantity (i.e. variable) depending on one or more other quantities. Thus we would speak about a variable z depends differentiably on y1,,yn, which in turn depend differentiably on variables x1,,xm. We would then write the chain rule as


The most general, and conceptually clear approach to the multi-variable chain is based on the notion of a differentiable mapping, with the Jacobian matrix of partial derivatives playing the role of generalized derivative. Let, Xm and Yn be open domains and let


be differentiable mappings. In essence, the symbol 𝐅 represents l functions of n variables each:


whereas 𝐆=(G1,,Gn) represents n functions of m variables each. The derivative of such mappings is no longer a function, but rather a matrix of partial derivatives, customarily called the Jacobian matrix. Thus

D𝐅=(D1F1DnF1D1FlDnFl)  D𝐆=(D1G1DmG1D1GnDmGn)

The chain rule now takes the same form as it did for functions of one variable:


albeit with matrix multiplicationMathworldPlanetmath taking the place of ordinary multiplication.

This form of the chain rule also generalizes quite nicely to the even more general setting where one is interested in describing the derivative of a composition of mappings between manifoldsMathworldPlanetmath.

Title chain rule (several variables)
Canonical name ChainRuleseveralVariables
Date of creation 2013-03-22 12:33:10
Last modified on 2013-03-22 12:33:10
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Theorem
Classification msc 26B12
Related topic ChainRule
Related topic JacobianMathworldPlanetmath
Related topic JacobianMatrix