chain rule (several variables)

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

th

variable.

Let

F

⁢

(

x

1

,

…

,

x

n

)

and

G

1

⁢

(

x

1

,

…

,

x

m

)

,

…

,

G

n

⁢

(

x

1

,

…

,

x

m

)

be

differentiable
functions

of several variables, and let

H

⁢

(

x

1

,

…

,

x

m

)

=

F

⁢

(

G

1

⁢

(

x

1

,

…

,

x

m

)

,

…

,

G

n

⁢

(

x

1

,

…

,

x

m

)

)

be the function determined by the

composition

of

F

with

G

1

,

…

,

G

n

The partial derivatives of

H

are given by

(

D

k

⁢

H

)

⁢

(

x

1

,

…

,

x

m

)

=

∑

i

=

1

n

(

D

i

⁢

F

)

⁢

(

G

1

⁢

(

x

1

,

…

,

x

m

)

,

…

)

⁢

(

D

k

⁢

G

i

)

⁢

(

x

1

,

…

,

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

.html

historical 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

,

…

,

y

n

,
which in turn depend differentiably on variables

x

1

,

…

,

x

m

. We
would then write the chain rule as

∂

⁡

z

∂

⁡

x

j

=

∑

i

=

1

n

∂

⁡

z

∂

⁡

y

i

⁢

∂

⁡

y

i

∂

⁡

x

j

,

j

=

1

,

…

⁢

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

⊂

ℝ

m

and

Y

⊂

ℝ

n

be open domains and let

𝐅

:

Y

→

ℝ

l

,

𝐆

:

X

→

Y

be differentiable mappings. In essence, the symbol

𝐅

represents

l

functions of

n

variables each:

𝐅

=

(

F

1

,

…

,

F

l

)

,

F

i

=

F

i

⁢

(

x

1

,

…

,

x

n

)

,

whereas

𝐆

=

(

G

1

,

…

,

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

⁢

𝐅

=

(

D

1

⁢

F

1

…

D

n

⁢

F

1

⋮

⋱

⋮

D

1

⁢

F

l

…

D

n

⁢

F

l

)

D

⁢

𝐆

=

(

D

1

⁢

G

1

…

D

m

⁢

G

1

⋮

⋱

⋮

D

1

⁢

G

n

…

D

m

⁢

G

n

)

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

D

⁢

(

𝐅

∘

𝐆

)

=

(

(

D

⁢

𝐅

)

∘

𝐆

)

⁢

(

D

⁢

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

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