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Topology is geometry without measuring. It concerns the properties of geometric figures that remain when concepts like length, area, and volume are disregarded. A good example of such a property is the number of holes in a surface. In the 1990s, Goodwillie formulated a topological calculus, which somewhat resembles the classical calculus taught in high school. This calculus reveals a number of very interesting topological phenomena. Arone and Ching later proved that, in certain cases, a chain rule holds for this calculus. In this dissertation, I prove a much more general form of this chain rule. This proof was found in collaboration with Thomas Blom.