chain rule (several variables)

The chain rule is a theorem of analysis that governs derivatives ofcomposed functions. The basic theorem is the chain rule for functionsof one variables (see here (http://planetmath.org/ChainRule)). This entryis devoted to the more general version involving functions of severalvariables and partial derivatives. Note: the symbol $D_{k}$ will beused to denote the partial derivative with respect to the $k^{\\text{th}}$ variable.

Let $F(x_{1},\\ldots,x_{n})$ and $G_{1}(x_{1},\\ldots,x_{m}),\\ldots,G_{n}(x_{1},\\ldots,x_{m})$ be differentiablefunctions of several variables, and let

H(x_{1},\\ldots,x_{m})=F(G_{1}(x_{1},\\ldots,x_{m}),\\ldots,G_{n}(x_{1},\\ldots,x_%{m}))

be the function determined by the composition of $F$ with $G_{1},\\ldots,G_{n}$ The partial derivatives of $H$ are given by

(D_{k}H)(x_{1},\\ldots,x_{m})=\\sum_{i=1}^{n}(D_{i}F)(G_{1}(x_{1},\\ldots,x_{m}),%\\ldots)(D_{k}G_{i})(x_{1},\\ldots,x_{m}).

The chain rule can be more compactly (albeit less precisely) expressedin terms of the Jacobi-Legendre partial derivative symbols(http://members.aol.com/jeff570/calculus.htmlhistorical note). Just as inthe Leibniz system, the basic idea is that of one quantity (i.e.variable) depending on one or more other quantities. Thus we wouldspeak about a variable $z$ depends differentiably on $y_{1},\\ldots,y_{n}$ ,which in turn depend differentiably on variables $x_{1},\\ldots,x_{m}$ . Wewould then write the chain rule as

\\frac{\\partial z}{\\partial x_{j}}=\\sum_{i=1}^{n}\\frac{\\partial z}{\\partial y_{%i}}\\frac{\\partial y_{i}}{\\partial x_{j}},\\qquad j=1,\\ldots m.

The most general, and conceptually clear approach to themulti-variable chain is based on the notion of a differentiablemapping, with the Jacobian matrix of partial derivatives playing therole of generalized derivative. Let, $X\\subset\\mathbb{R}^{m}$ and $Y\\subset\\mathbb{R}^{n}$ be open domains and let

\\mathbf{F}:Y\\rightarrow\\mathbb{R}^{l},\\qquad\\mathbf{G}:X\\rightarrow Y

be differentiable mappings. In essence, the symbol $\\mathbf{F}$ represents $l$ functions of $n$ variables each:

\\mathbf{F}=(F_{1},\\ldots,F_{l}),\\qquad F_{i}=F_{i}(x_{1},\\ldots,x_{n}),

whereas $\\mathbf{G}=(G_{1},\\ldots,G_{n})$ represents $n$ functions of $m$ variables each.The derivative of such mappings is no longer a function, but rather amatrix of partial derivatives, customarily called the Jacobian matrix.Thus

D\\mathbf{F}=\\begin{pmatrix}D_{1}F_{1}&\\ldots&D_{n}F_{1}\\\\\\vdots&\\ddots&\\vdots\\\\D_{1}F_{l}&\\ldots&D_{n}F_{l}\\end{pmatrix}\\qquad D\\mathbf{G}=\\begin{pmatrix}D_{%1}G_{1}&\\ldots&D_{m}G_{1}\\\\\\vdots&\\ddots&\\vdots\\\\D_{1}G_{n}&\\ldots&D_{m}G_{n}\\end{pmatrix}

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

D(\\mathbf{F}\\circ\\mathbf{G})=((D\\mathbf{F})\\circ\\mathbf{G})\\,(D\\mathbf{G}),

albeit with matrixmultiplication taking the place of ordinary multiplication.

This form of the chain rule also generalizes quite nicely to the evenmore general setting where one is interested in describing thederivative of a composition of mappings between manifolds.