Much of the education research on implicit differentiation and related rates treats the topic of differentiating equations as an unproblematic application of the chain rule. This paper instead problematizes the legitimacy of this procedure. It develops a conceptual analysis aimed at exploring how a student might come to understand when and why one can differentiate both sides of an equation and elucidates the central role functions play. The four steps developed in this conceptual analysis are then used to analyze the struggles of a calculus student.