Students’ ability to correctly apply differentiation rules to structurally different functions

Abstract

The derivative concept is studied in first-year university mathematics. In this study, we focused on students’ ability to correctly apply the rules for derivatives of functions with the different structures that they encounter in their university studies. This was done by investigating the online responses of first-year students at the University of KwaZulu-Natal to online quizzes that contributed to their assessment. Based on this investigation, we then interviewed eight students to gain an insight into the thinking behind their responses. We report on the analysis of students’ responses to five items on the online quizzes based on the derivative concept. The categories in which those items were based are: condition for existence of derivative at a point; rules for derivatives of standard functions; application of chain rule to different function structures; the application of multiple rules; and application of derivatives to optimise a function. Our findings indicate that students had difficulty in detecting that multiple rules for derivatives were required to differentiate certain types of functions represented in symbolic form. Furthermore, students had difficulty in finding the derivative of a function when more than one application of the chain rule was required. However, there were students who had the ability to apply the rules for derivatives of functions without difficulty. In particular, most of the students were able to correctly recall the differentiation rules for functions with standard structures f(x)=xⁿ, h(x)=e^kx and y=[g(x)]ⁿ, n ≠ 0 and k is a non-zero constant. Students were also able to correctly apply the chain rule to an exponential function with base e, raised to 4x. The majority of students were able to correctly apply the chain rule together with differentiation rules for logarithmic and exponential (with bases a >1) function structures, and function structures that required the application of the product rule together with the chain rule. Most of the students were able to apply derivatives to optimise a function.

Significance:

• A significant percentage of students who took online quizzes experienced difficulties with applying multiple differentiation rules in the context of a single function. The difficulties stemmed from their inability to detect from the structure of the function which rules should be applied and also the order in which those relevant rules should be applied.