12.7.2 The Derivative Of The Inverse Function
Example 12.7.6 Let f : U → V where U and V are open sets in ℝnand f is one to one and onto.
Suppose also that f and f−1 are both differentiable. How are Df−1 and Df related?
This can be done as follows. From the assumptions, x = f−1
Then by Example 12.3.6
on Page 595 DI
. By the chain rule,
Letting y = f
, the second yields
This is equivalent to
This is just like a similar situation for functions of one variable. Remember
In terms of the repeated index summation convention, suppose y = f
. Then the
above can be written as