Introductory Mathematics and Statistics for Islamic Finance, + Website

Chapter 4Partial Derivatives

Economic and financial relations feature interdependence of many variables. For instance, the utility function of a consumer may include many products and is written as

(4.1)

where , , and denote the quantities of oranges, tomatoes, and grape juice, respectively, consumed by the consumer. In economics, we are interested in how a dependent variable responds when only one variable in the relationship changes; the assumption of maintaining all other fixed variables is called the ceteris paribus condition. In this case we may be interested in computing the change in utility due to a change in the consumption of oranges, keeping the consumption of tomatoes and grape juice fixed. This rate of change is called the partial derivative of utility in respect to a change in oranges consumed. It is denoted as to indicate it is a partial derivative. We call it marginal utility of oranges.

Likewise, any production process involves the interplay of many variables. Production ...