The Derivative of a Function
Starting from the problem to define the tangent to the graph of a function, we introduce the derivative of a function. Two points on the graph can always be joined by a secant, which is a good model for the tangent whenever these points are close to each other. In a limiting process, the secant (discrete model) is replaced by the tangent (continuous model). Differential calculus, which is based on this limiting process, has become one of the most important building blocks of mathematical modelling.
In this section we discuss the derivative of important elementary functions as well as general differentiation rules. Thanks to the meticulous implementation of these rules, expert systems such as maple have become helpful tools in mathematical analysis. Furthermore, we will discuss the interpretation of the derivative as linear approximation and as rate of change. These interpretations form the basis of numerous applications in science and engineering.
The concept of the numerical derivative follows the opposite direction. The continuous model is discretised and the derivative is replaced by a difference quotient. We carry out a detailed error analysis which allows us to find an optimal approximation. Further, we will illustrate the relevance of symmetry in numerical procedures.