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.