ABSTRACT
Differentiation The derivative of a function, as presented in standard courses in differential calculus, is usually tied up with the notion of the slope of the tangent to the curve which represents the function, or with the rate of change of the function. These interpretations are, in fact, the reason differentiation was introduced. Once analytic geometry was developed in the early part of the seventeenth century, many geometric and physical relations were expressed by algebraic relations. As it turned out, the slope of the geometric curve which represents the equation y = f(x) and the rate of change of the physical quantity y with respect to x are one and the same, expressed by the concept of the derivative. René Descartes (1596-1650) is credited with the introduction of analytic geometry, which was shortly followed by the development of differential and integral calculus at the hands of Isaac Newton (16421727) and Gottfried Wilhelm Leibnitz (1646-1716). Thus, in the course of about forty years, from 1630 to 1670, mathematics went through a phase of remarkable expansion unparalleled in its entire recorded history. In this chapter we shall be mainly interested in the derivative from
an analytical point of view, as a development of the concepts which we have presented so far. Its physical interpretations, important as they are, lie outside the scope of this treatment. Integration will be presented along the lines of Bernhard Riemann (1826-1866) in Chapter 8.
