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-IRead More

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