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# Chain rules Notes | EduRev

## : Chain rules Notes | EduRev

``` Page 1

32.1 Chain rules

Recall that for a differentiable function of one-variable, if is also a differentiable
function of then the composite function is also a differentiable function of and
Similar results hold for functions of several variables.
We state next another such rule, the proof of which is similar to the Chain rule-I, and is left as an
exercise.
32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional
derivatives
Lecture 32 :  Chain rules [Section 32.1]
Objectives
In this section you will learn the following :
Chain rules which help to compute partial derivatives of composite functions.

Page 2

32.1 Chain rules

Recall that for a differentiable function of one-variable, if is also a differentiable
function of then the composite function is also a differentiable function of and
Similar results hold for functions of several variables.
We state next another such rule, the proof of which is similar to the Chain rule-I, and is left as an
exercise.
32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional
derivatives
Lecture 32 :  Chain rules [Section 32.1]
Objectives
In this section you will learn the following :
Chain rules which help to compute partial derivatives of composite functions.

and

Functionally, this is also written as

32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
and

Functionally, this is also written as
Proof

Let
Then, by differentiability of , we have for ,
Thus for
--------(28)
Since are continuous, as Hence, as , it follows from (28)
that is differentiable and

32.1.2Theorem (Chain rule-II):

Page 3

32.1 Chain rules

Recall that for a differentiable function of one-variable, if is also a differentiable
function of then the composite function is also a differentiable function of and
Similar results hold for functions of several variables.
We state next another such rule, the proof of which is similar to the Chain rule-I, and is left as an
exercise.
32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional
derivatives
Lecture 32 :  Chain rules [Section 32.1]
Objectives
In this section you will learn the following :
Chain rules which help to compute partial derivatives of composite functions.

and

Functionally, this is also written as

32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
and

Functionally, this is also written as
Proof

Let
Then, by differentiability of , we have for ,
Thus for
--------(28)
Since are continuous, as Hence, as , it follows from (28)
that is differentiable and

32.1.2Theorem (Chain rule-II):

Let and be differentiable at Let and
be functions such that
If exist at then the composite function
has partial derivatives and given by
Symbolically,
32.1.3Examples:
(i) Let

and

Let
Then, by chain rule, is differentiable for all and we have
(ii) Let

and

For
we have by chain rule,
,

Page 4

32.1 Chain rules

Recall that for a differentiable function of one-variable, if is also a differentiable
function of then the composite function is also a differentiable function of and
Similar results hold for functions of several variables.
We state next another such rule, the proof of which is similar to the Chain rule-I, and is left as an
exercise.
32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional
derivatives
Lecture 32 :  Chain rules [Section 32.1]
Objectives
In this section you will learn the following :
Chain rules which help to compute partial derivatives of composite functions.

and

Functionally, this is also written as

32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
and

Functionally, this is also written as
Proof

Let
Then, by differentiability of , we have for ,
Thus for
--------(28)
Since are continuous, as Hence, as , it follows from (28)
that is differentiable and

32.1.2Theorem (Chain rule-II):

Let and be differentiable at Let and
be functions such that
If exist at then the composite function
has partial derivatives and given by
Symbolically,
32.1.3Examples:
(i) Let

and

Let
Then, by chain rule, is differentiable for all and we have
(ii) Let

and

For
we have by chain rule,
,

(iii) Let

Then, exist everywhere and
Hence, if
then
However, since
we have This does not contradict Chain Rule as is not differentiable (in fact not even
continuous ) at
32.1.4Remark:
Chain rule extends to functions of three or more variables as above for functions of two variables.

Page 5

32.1 Chain rules

Recall that for a differentiable function of one-variable, if is also a differentiable
function of then the composite function is also a differentiable function of and
Similar results hold for functions of several variables.
We state next another such rule, the proof of which is similar to the Chain rule-I, and is left as an
exercise.
32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
Module 11 :   Partial derivatives, Chain rules,  Implicit differentiation, Gradient, Directional
derivatives
Lecture 32 :  Chain rules [Section 32.1]
Objectives
In this section you will learn the following :
Chain rules which help to compute partial derivatives of composite functions.

and

Functionally, this is also written as

32.1.1Theorem (Chain rule-I:)

Let and be differentiable at Let and

be functions such that

If x, y are both differentiable at then the composite function
given by
is differentiable at
and

Functionally, this is also written as
Proof

Let
Then, by differentiability of , we have for ,
Thus for
--------(28)
Since are continuous, as Hence, as , it follows from (28)
that is differentiable and

32.1.2Theorem (Chain rule-II):

Let and be differentiable at Let and
be functions such that
If exist at then the composite function
has partial derivatives and given by
Symbolically,
32.1.3Examples:
(i) Let

and

Let
Then, by chain rule, is differentiable for all and we have
(ii) Let

and

For
we have by chain rule,
,

(iii) Let

Then, exist everywhere and
Hence, if
then
However, since
we have This does not contradict Chain Rule as is not differentiable (in fact not even
continuous ) at
32.1.4Remark:
Chain rule extends to functions of three or more variables as above for functions of two variables.

Figure 1. Chain rule-I

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