proof of chain rule (several variables)

proof of chain rule (several variables)

We first consider the case

m

=

1

i.e.

G

:

I

→

ℝ

n

where

I

⊂

ℝ

is a neighbourhood of a point

x

0

∈

ℝ

and

F

:

U

⊂

ℝ

n

→

ℝ

is defined on a neighbourhood

U

of

y

0

=

G

⁢

(

x

0

)

such that

G

⁢

(

I

)

⊂

U

. We suppose that both

G

is

differentiable

at the point

x

0

and

F

is differentiable in

y

0

. We want to compute the

derivative

of the compound

function

H

⁢

(

x

)

=

F

⁢

(

G

⁢

(

x

)

)

at

x

=

x

0

.

By the definition of derivative (using

Landau notation

) we have

F

⁢

(

y

0

+

k

)

=

F

⁢

(

y

0

)

+

D

⁢

F

⁢

(

y

0

)

⁢

k

+

o

⁢

(

|

k

|

)

.

Choose any

h

≠

0

such that

x

0

+

h

∈

I

and set

k

=

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

to obtain

H

⁢

(

x

0

+

h

)

-

H

⁢

(

x

0

)

h

=

F

⁢

(

G

⁢

(

x

0

+

h

)

)

-

F

⁢

(

G

⁢

(

x

0

)

)

h

=

F

⁢

(

G

⁢

(

x

0

)

+

k

)

-

F

⁢

(

G

⁢

(

x

0

)

)

h

=

F

⁢

(

y

0

+

k

)

-

F

⁢

(

y

0

)

h

=

D

⁢

F

⁢

(

y

0

)

⁢

(

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

)

+

o

⁢

(

|

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

|

)

h

=

D

⁢

F

⁢

(

y

0

)

⁢

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

h

+

o

⁢

(

|

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

|

)

h

.

Letting

h

→

0

the first term of the sum converges to

D

⁢

F

⁢

(

y

0

)

⁢

G

′

⁢

(

x

0

)

hence
we want to prove that the second term converges to

0

.
Indeed we have

|

o

⁢

(

|

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

|

)

h

|

=

|

o

⁢

(

|

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

|

)

|

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

|

|

⋅

|

G

⁢

(

x

0

+

h

)

-

G

⁢

(

x

0

)

h

|

.

By the definition of

o

⁢

(

⋅

)

the first

fraction

tends to

0

, while
the second fraction tends to the

absolute value

of

G

′

⁢

(

x

0

)

. Thus the product
tends to

0

, as needed.

Consider now the general case

G

:

V

⊂

ℝ

m

→

U

⊂

ℝ

n

.
Given

v

∈

ℝ

m

we are going to compute the

directional derivative

∂

⁡

F

∘

G

∂

⁡

v

⁢

(

x

0

)

=

d

⁢

F

∘

g

d

⁢

t

⁢

(

0

)

where

g

⁢

(

t

)

=

G

⁢

(

x

0

+

t

⁢

v

)

is a function of a single variable

t

∈

ℝ

. Thus
we fall back to the previous case and we find that

∂

⁡

F

∘

G

∂

⁡

v

(

x

0

)

=

D

F

(

G

(

x

0

)

)

g

′

(

0

)

.

=

D

F

(

G

(

x

0

)

)

∂

⁡

G

∂

⁡

v

(

x

0

)

In particular when

v

=

e

k

is the

k

-th

coordinate vector

, we find

g

′

⁢

(

0

)

=

D

x

k

⁢

F

∘

G

⁢

(

x

0

)

=

D

⁢

F

⁢

(

G

⁢

(

x

0

)

)

⁢

D

x

k

⁢

G

⁢

(

x

0

)

=

∑

i

=

1

n

D

y

i

⁢

G

⁢

(

x

0

)

⁢

D

x

k

⁢

G

i

⁢

(

x

0

)

which can be compactly written

D

⁢

F

∘

G

⁢

(

x

0

)

=

D

⁢

F

⁢

(

G

⁢

(

x

0

)

)

⁢

D

⁢

G

⁢

(

x

0

)

.

Title

proof of chain rule

(several variables)

Canonical name

ProofOfChainRuleseveralVariables

Date of creation

2013-03-22 16:05:07

Last modified on

2013-03-22 16:05:07

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

6

Author

paolini (1187)

Entry type

Proof

Classification

msc 26B12

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