Polyfold theory is built on two new fundamental analysis concepts, and this talk is focused of the first: the sc-calculus. We discuss how in general the action of a finite dimensional smooth reparametrization group on typical Banach spaces of maps is not smooth (in fact, not even differentiable), and then introduce sc-Banach spaces and the notion of sc-differentiability. Two key results of this talk are that the action of reparametrization is sc-smooth, and for sc-differentiable functions the chain rule holds, so that many constructions in classical differential geometry functorially extend to sc-differentiable geometry. [Related literature: Sections 2.2 and 4.2 of Polyfolds: A First and Second Look.]