Module 3 : Differentiation and Mean Value Theorems
Lecture 8 : Chain Rule [Section 8.1]
8.1.11
Implicit Differentiation :
Sometimes, the relation between the independent variable and the dependent variable is not explicitly given a function , but is given as a relation . For example, the relation , can be written implicitly as . However, the relation , does not allow us to represent explicitly as a function of . In fact, it represents more than one function. The representation is called implicit representation of the function . The question one wants to answer is the following: When can we compute from the implicit relation , without requiring to compute in terms of explicitly? To provide a complete answer to this we need a theorem from advanced calculus called “Implicit Function Theorem”, which gives conditions under which an implicit equation represents an explicit function , and is differentiable. Further, it ensures that to find one can differentiate using the rules of differentiation and solve it for . This theorem is stronger than the theorem on the derivatives of inverse function.
8.1.12
Example:
Consider the relation which can be written as . By the implicit function theorem and using chain the rule, we get
, thus i.e, .
Of course, the question arises: ‘which is the right derivative?'. For that, we observe that represents 2-different functions : and .
and the dependent variable 
is not explicitly given a function 
, but is given as a relation 
. For example, the relation 
, can be written implicitly as 
. However, the relation 
, does not allow us to represent 
explicitly as a function of 
. In fact, it represents more than one function. The representation 
is called implicit representation of the function . The question one wants to answer is the following: When can we compute 
from the implicit relation 
, without requiring to compute 
in terms of explicitly? To provide a complete answer to this we need a theorem from advanced calculus called “Implicit Function Theorem”, which gives conditions under which an implicit equation 
represents an explicit function 
, and is differentiable. Further, it ensures that to find 
one can differentiate 
using the rules of differentiation and solve it for 
. This theorem is stronger than the theorem on the derivatives of inverse function. 
which can be written as 
. By the implicit function theorem and using chain the rule, we get 
, thus 
i.e, 
. 
represents 2-different functions : 
and 
.
