As many calculus instructors realize, it is one thing for students to learn derivative techniques or rules in class. It is quite another for students to know when to apply — or not to apply — a specific rule. How often has a student told you, “I could follow the in-class discussion of the problems but I got lost as soon as I had to do the problems on my own?”
I have created an assignment to assist my students with internalizing those characteristics of functions that determine which differentiation technique(s) to use. The assignment is based on standard end-of- chapter lists of review exercises.
Instead of giving my students lists of functions to differentiate using various rules, I reverse the instructions. My students must decide which characteristics a function should possess in order to use a specific differentiation rule. From the review exercises, my students must choose two functions to differentiate in each of the following categories: Product Rule, Quotient Rule, Chain Rule, Exponential Rule for base e, Exponential Rule for bases other than e, Natural Log Rule, and any of the Trigonometric Rules. Then, students must comment on common attributes possessed by the original functions in each category. No function may be used in more than one category and all work must be shown for maximum credit.
The end-of-chapter list of review exercises blends the differentiation rules, giving students no hints as to which rule is appropriate when. Since knowing the derivative of a function often does not give a clue as to the differentiation technique used, technology is not necessarily an advantage.
Students improve their judgment of which differentiation rules to use (or not to use). Typical student reaction to this assignment is “This assignment helped me see characteristics of functions. I now know and understand which rules to use. I can see distinguishing characteristics in functions.”
A similar assignment can be designed for integration techniques.
Time spent: about 20 minutes to create the assignment listing your own derivative techniques.
Time saved: about 2 minutes, on average, for every exam or other differentiation assignment you grade.
Pam Crawford is an Associate Professor and Chair of the Mathematics Department at Jacksonville University in Jacksonville, Florida.