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
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Read More
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!