The Fill-the-Cell Grid: A Uniform Setup for Four Derivative and Integral Rule

A practitioner's explainer describing a single visual routine — a small grid filled in cell by cell — that gives the product rule, the quotient rule, the chain rule, and single-application integration by parts the same setup shape. The aim is to lower the point at which first-year calculus students stall: not the concept, but where to write the first piece down. The document works five examples across the four rules and describes a uniform "fill-the-cell" setup, an empty-box separation of operation from content, and a marked "start" cell that fixes the quotient rule's subtraction order. The approach is offered as a consolidation, not an invention. The integration-by-parts grid sits in an established tabular lineage (Folley 1947; Murty 1980; Horowitz 1990; Rock 2016); the product, quotient, and chain-rule grids use a cross-and-box visual that is common instructional practice. The contribution is the uniform setup and its novice-facing description, not the underlying visual idea. The document presents a teaching setup only and reports no student-outcome data.