calculus in normed vector spaces is the basis for several areas of mathematics and physics, but it is not a topic that is often covered with very much care or detail. this talk's focus is the theory of differentiation in normed vector spaces, more specifically the gateaux and fr\'{e}chet derivatives. towards the end, we shall cover an infinite-dimensional taylor's theorem, and we shall likely get to discuss some applications. plus, there will be plenty of examples throughout! this talk is part i of a (likely) three- or four-part series, with future topics including integration and complex analysis.