# chain rule (several variables)

The chain rule^{} 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 (http://planetmath.org/ChainRule)). This entry
is devoted to the more general version involving functions of several
variables and partial derivatives^{}. Note: the symbol ${D}_{k}$ will be
used to denote the partial derivative with respect to the ${k}^{\text{th}}$
variable.

Let $F({x}_{1},\mathrm{\dots},{x}_{n})$ and ${G}_{1}({x}_{1},\mathrm{\dots},{x}_{m}),\mathrm{\dots},{G}_{n}({x}_{1},\mathrm{\dots},{x}_{m})$ be differentiable functions of several variables, and let

$$H({x}_{1},\mathrm{\dots},{x}_{m})=F({G}_{1}({x}_{1},\mathrm{\dots},{x}_{m}),\mathrm{\dots},{G}_{n}({x}_{1},\mathrm{\dots},{x}_{m}))$$ |

be the function determined by the composition^{} of $F$ with
${G}_{1},\mathrm{\dots},{G}_{n}$
The partial derivatives of $H$ are given by

$$({D}_{k}H)({x}_{1},\mathrm{\dots},{x}_{m})=\sum _{i=1}^{n}({D}_{i}F)({G}_{1}({x}_{1},\mathrm{\dots},{x}_{m}),\mathrm{\dots})({D}_{k}{G}_{i})({x}_{1},\mathrm{\dots},{x}_{m}).$$ |

The chain rule can be more compactly (albeit less precisely) expressed in terms of the Jacobi-Legendre partial derivative symbols (http://members.aol.com/jeff570/calculus.htmlhistorical 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 ${y}_{1},\mathrm{\dots},{y}_{n}$, which in turn depend differentiably on variables ${x}_{1},\mathrm{\dots},{x}_{m}$. We would then write the chain rule as

$$\frac{\partial z}{\partial {x}_{j}}=\sum _{i=1}^{n}\frac{\partial z}{\partial {y}_{i}}\frac{\partial {y}_{i}}{\partial {x}_{j}},j=1,\mathrm{\dots}m.$$ |

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, $X\subset {\mathbb{R}}^{m}$ and $Y\subset {\mathbb{R}}^{n}$ be open domains and let

$$\mathbf{F}:Y\to {\mathbb{R}}^{l},\mathbf{G}:X\to Y$$ |

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

$$\mathbf{F}=({F}_{1},\mathrm{\dots},{F}_{l}),{F}_{i}={F}_{i}({x}_{1},\mathrm{\dots},{x}_{n}),$$ |

whereas $\mathbf{G}=({G}_{1},\mathrm{\dots},{G}_{n})$ 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

$$D\mathbf{F}=\left(\begin{array}{ccc}\hfill {D}_{1}{F}_{1}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {D}_{n}{F}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {D}_{1}{F}_{l}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {D}_{n}{F}_{l}\hfill \end{array}\right)\mathit{\hspace{1em}\hspace{1em}}D\mathbf{G}=\left(\begin{array}{ccc}\hfill {D}_{1}{G}_{1}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {D}_{m}{G}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {D}_{1}{G}_{n}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {D}_{m}{G}_{n}\hfill \end{array}\right)$$ |

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

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

albeit with matrix
multiplication^{} 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 manifolds^{}.

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 | Jacobian^{} |

Related topic | JacobianMatrix |