Textbook
Keisler's textbook is based on Robinson's construction of the hyperreal numbers. Because this is not a subject widely known Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors which covers the foundational material in more depth.
Keisler defines all basic notions of the calculus such as continuity, derivative, and integral using infinitesimals. The usual definitions in terms of ε-δ techniques are provided at the end of Chapter 5 to enable a transition to a standard sequence.
In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hyperreal numbers infinitely close to each other. Similarly, an infinite-resolution telescope is used to represent infinite numbers.
When one examines a curve, say the graph of ƒ, under a magnifying glass, its curvature decreases proportionally to the magnification power of the lens. Similarly, an infinite-magnification microscope will transform an infinitesimal arc of a graph of ƒ, into a straight line, up to an infinitesimal error (only visible by applying a higher-magnification "microscope"). The derivative of ƒ is then the (standard part of the) slope of that line. Thus the microscope is used as a device in explaining the derivative.
Read more about this topic: Elementary Calculus: An Infinitesimal Approach