Original, rigorous, and lively, this text offers a concise approach to classical and contemporary topics in differential calculus. Based on courses conducted by the author at the Université Pierre et Marie Curie, it encourages readers to pursue the subject in greater depth. The calculus is presented in a Banach space setting, covering: - Vector fields - One-parameter groups of diffeomorphisms - The Morse-Palais lemma - Differentiable submanifolds The treatment also examines applications to differential equations and the calculus of variables. For upper-level undergraduates and graduate students of analysis.