The present small volume is intended to form a sound introduction to a study of the Differential Calculus suitable for the beginner. It does not therefore aim at completeness, but rather at the omission of all portions which are usually considered best left for a later reading. At the same time it has been constructed to include those parts of the subject prescribed in Schedule I. of the Regulations for the Mathematical Tripos Examination for the reading of students for Mathematical Honours in the University of Cambridge. Particular attention has been given to the examples which are freely interspersed throughout the text. For the most part they are of the simplest kind, requiring but little analytical skill. Yet it is hoped they will prove sufficient to give practice in the processes they are intended to illustrate. It is assumed that in commencing to work at the Differential Calculus the student possesses a fair knowledge of Algebra as far as the Exponential and Logarithmic Theorems; of Trigonometry as far as Demoivre’s Theorem, and of the rudiments of Cartesian Geometry as far as the equations of the several Conic Sections in their simplest forms