The authors, affiliated with Utrecht University, present a two-volume work on the theory of vector-valued functions in multidimensional Euclidean space. This volume is devoted to differentiation, with material on differentiable functions on Rn and differentiable manifolds embedded in Rn. Chapters cover continuity, differentiation, inverse function and implicit function theorems, manifolds, and tangent spaces. Chapter exercises illustrate a variety of applications in mathematics and physics. The book can be used as a text for a course for students who wish to go on to more advanced study, as a source of problems for a seminar, or for self study.