Solution 19 (a).

See text and/or instructor's solution manual.

Answer.   It is easy to show that    for all positive integers  n.  Do so via mathematical induction.

Solution.   Start with   and use the rules of differentiation to get

                    .

So that     is true.

The next derivative is

                    .

We actually need to use mathematical induction to establish the general formula for the derivative:

                    .

Assume that for some  ,  we have .

Then   

                       

Therefore, by mathematical induction      is true for all   .  

Now we can evaluate    and obtain   for all positive integers  n.  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell