Extending the Derivative Concept: A Coherent, Quantitative Approach to the Chain Rule, Implicit Differentiation, and Related Rates

The “rate of change” or “ratio of infinitesimal changes” meanings for derivatives are crucial to productively using calculus across STEM disciplines. While much research has examined the basic derivative concept itself, less work has examined how the derivative might be extended across a unit on derivatives in ways that are compatible with this rate/ratio meaning. In this paper, we propose a hypothetical learning trajectory (HLT) for coherently teaching the derivative extensions of the chain rule, implicit differentiation, and related rates under this quantitative paradigm. The HLT extends the covariational rate/ratio meaning for the basic derivative to a multivariational meaning in these three extension topics. We also present the results of a small-scale teaching experiment meant to test the plausibility of the HLT. Our results suggest the students developed the nested $$\:x\to\:y\to\:z$$ relationship, and used it to construct the rate1 × rate2 = rate3 structure, which was often formulated through “ratios” of small changes. The results also imply that coherence was achieved in these lessons in that the students saw these three topics as related and connected under this same structure.