Definition:

Let be an interval and . We say is twice differentiable at if is differentiable on for some and the derivative function is differentiable at . In that case we define the second order derivative of at to be

                                         ,

the derivative of the derivative function. It is also denoted by

                                      
The concept of differentiability and the derivative of at , denoted by , can be defined similarly:
                                       

If exists for every , we say is infinitely differentiable at .

(i) Consider

    Then is differentiable for every and and