Module 3 : Differentiation and Mean Value Theorems
Lecture 8 : Chain Rule [Section 8.1]
8.1.7
Example :
Let and Then,
Since for , we get , for .
In this case, in fact as a function of is given by , and hence
8.1.8
Note :
In general, the graph of a curve, may not arise as the graph of a function. For example
defines a curve in the plane, called prolete cycloid. Even though at the point ,
and we have ,
however, this is not the derivative of any function .
We saw in the lecture 6 that the inverse of a one-one continuous function is also continuous. It is natural to ask the question: When is the inverse of a one one differentiable function also differentiable? The answer is the next theorem.
and 
Then, 
for 
, we get 
, for 
. 
as a function of 
is given by 
, and hence 
may not arise as the graph of a function. For example 
, 
, 
. 
