An examination of calculus students’ quantitative and covariational reasoning in the context of differentials

Abstract This study reports on students’ quantitative reasoning, covariational reasoning, functional knowledge and calculational knowledge in the context of working with differentials. Analysis of work and verbal responses provided by 10 calculus students when working on four tasks revealed that nearly all the students drew and used diagrams to support their quantitative reasoning and that mathematizing application problems involving differentials was straightforward for these students. Additionally, most of the students exhibited weak covariational reasoning abilities. Recognizing $dy={f}^{\prime }(x) dx$ as a function was straightforward for all the students, although determining the dependent variable, independent variables and domain of $x$ and $dx$ was problematic for the students. Findings of this study indicate that students’ poor calculational knowledge of differentials stems from a lack of facility with various rules of differentiation. Implications for instruction and directions for future research are included.