Module 3 : Differentiation and Mean Value Theorems
Lecture 8 : Chain Rule [Section 8.1]
8.1.4
Example (Derivative of the exponential function):
In example 3.1.6 (iii) we observed that the function is a bijective function which is differentiable at every point . Its inverse function is called the exponential function and is denoted by .
Since has derivative for every , the exponential function is differentiable. If , then
Hence, is its own derivative for every .
8.1.5
Example (Derivative of general power function):
Let and for every define
Note that where By chain rule, is differentiable at every point and
We give below some applications of the chain rule.
is a bijective function which is differentiable at every point 
. Its inverse function is called the 
. 
has derivative 
for every 
, the exponential function is differentiable. If 
, then 
is its own derivative for every 
. 
and for every 
define 
where 
By chain rule, 
is differentiable at every point 
and 
