This 36-page Microworld is an Active excursion into one of the most basic concepts of the Calculus: the Derivative. It develops this idea with many simulations, illustrations, examples and exercises. Throughout the book, readers may ask their own questions and study their own examples.

It begins by illustrating the graphical representation of derivative as slope with lively animations: physical notions of speed, steepness of ascent, and growth of geometric objects. Here, the author describes in an informal way, the common sense ideas behind the constructions.

Next, it defines the derivative in a formal way in terms of limits, and illustrates this limiting process for two-sided and left- and right-hand derivatives. After that, the book goes on to explore the relationships between continuity and differentiability. In particular, it studies some interesting counterexamples to the erroneous hypothesis that continuity implies differentiability.

Moving into the algebraic content of the subject, the next section presents a number of algebraic rules for calculating derivatives of functions. This begins with the usual formulas for specific classes of functions: polynomial functions, trigonometric, exponential, logarithmic, etc. But then it backs up and explores where these rules come from with several exploratory exercises that offer a way to visualize the rules and to compare the algebraic procedures with their pictorial and graphical correspondents.

Having stated and illustrated the basic differentiation rules, the Chapter on "Arithmetic of Derivatives" puts it all together in an interesting new way. That chapter explores step by step such rules as "The Sum Rule", "The Product Rule", "Quotient Rule" by allowing the student to supply her own examples and then giving a step-by-step application of the relevant rule to calculate the derivative. In this way, the student can see the rule applied to problems that she supplies, and therefore, the student will be more likely to understand the calculation.

Next, follows a discussion of the "Chain Rule" within the context of the same pedagogic strategy outlined above. This is very rich, because the student can easily supply examples whose calculation will lead to surprises. The calculations are always explained step by step, and there is an unbounded set of possibilities...

After presenting a "general strategy" for differentiation, the book moves to its Center: the Exercises. The Exercise section can be used to generate thousands of problems that will give practice in solving the derivative problems based on the basic derivative laws listed earlier. When the student clicks on any of the four buttons (Sum or Difference of functions, Product of Functions, Quotient of Functions, or Composition of Functions), a function of the listed type is created and printed. The student attempts to solve the problem by hand or with the Symbolic Calculator which is available from most pages. Students may check their answer using the visual and symbolic "Derivative Checker."