Module 3 : Differentiation and Mean Value Theorems
Lecture 8 : Chain Rule [Section 8.1]
8.1.9
Theorem (Derivative of the inverse function):
Let be an open interval and be a one-one differentiable function. Let be the range of and be the inverse function. Then, is differentiable at a point for , such that, and in that case
8.1.10
Example (differentiation of the n-th root function):
For , the function defined by is a one-one differentiable function with for . Thus, the inverse function is differentiable for every and its derivative at is given by
Consequently (by the Chain Rule), if is any rational number, then defines a differentiable function on and
for
Click here to see an interactive visualization (Java) : Derivative of the inverse function : Applet 8.2
be an open interval and 
be a one-one differentiable function. Let 
be the range of 
and 
be the inverse function. Then, 
is differentiable at a point 
for 
, such that, 
and in that case
, the function 
defined by 
is a one-one differentiable function with 
for 
. Thus, the inverse function 
is differentiable for every 
and its derivative at 
is given by 
is any rational number, then 
defines a differentiable function on 
and 
for 
