Exercise 7.  Find the derivatives of the following and state where they are defined.  

7 (c).   .  

Solution 7 (c).

See text and/or instructor's solution manual.

Answer.   ,   is valid for  ,  where  k  is an integer.

Solution.   Use the chain rule and obtain  

                      

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  When we apply the "rules for differentiation" we miss the point that we are working with functions of a complex variable.

Would we want to use the "Cauchy-Riemann" equations for finding a derivative of a complex function?  Would it be tractable?

These details are given below with the assistance of Mathematica.

First, expand the given function    in its    form:

Third, expand the derivative    we found in its    form:

Fourth, observe that the two versions of the derivative are the same.

Remark.  Without the assistance of computer algebra software we might not have given this demonstration.  

This solution is complements of the authors.

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