# Chapter 3Differentiation and Integration of Functions

Differentiation and integration of a function are the main analytical tools in Islamic finance. Differentiation and integration are related. The derivative of a function, if integrated, returns the function itself; the integral of a function, if differentiated, returns the function itself. The process of finding a derivative is called *differentiation*. The reverse process is called *integration*. Basically, integration is a summation operation; it is summing areas or volumes of the form , where is a small interval.

This chapter covers the techniques of differentiation and integration, introducing the notion of derivatives of functions, the notions of maxima and minima of a function, Taylor expansion for a differentiable function, the mean-value theorem, integration of functions, and presents applications in Islamic finance related to duration and convexity of a sukuk.^{1}

## Differentiation

Derivatives of functions are essential in economics and finance. We have to compute derivatives of functions such as marginal utility, marginal productivity, or marginal conditions in optimization theory. The derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity ...