Take 15 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
If you have more questions, feel to ask then on the forum.
Question
a) Prove that application
(defined in a previous exercise) is differentiable on
and calculate its derivative.
b) Prove that application
(defined in a previouis exercise) is differentiable on
and calculate its derivative.
a) et b) Apply the formula
and remember that:
Application
has for derivative the function
hence the derivative of application
is non zero on interval
There results from the theorem of the differentiation of the inverse of an application that application
is differentiable on interval
and
We then infer from formula
since applications
and
are each other's inverse.
b) Application
as for derivative the function
hence the derivative of application
is non zero on interval
There results from the theorem of the differentiation of the inverse of an application that application
is differentiable on interval
and:
We then infer from formula
since applications
and
are each other's inverse.
