Chapters 11 and 12 described the derivative of a function and how derivatives may be used to answer questions in business and economics. From a basic set of rules, we were able to calculate a huge number of derivatives. In this section, we’ll reverse this process. Instead of starting with a function and taking its derivative, we’ll start with the derivative and find the function it came from.
In earlier sections, we learned how to take the derivatives of sums, differences, products and quotients of two functions. These operations form four of the basic operations we can perform on functions. In this section, we complete derivatives on function operations by learning how to differentiate functions that can be written as a composition.
We’ll start by reviewing compositions and then present the Chain Rule for Derivatives. This rule will allow us to compute derivatives of compositions. Finally, we’ll combine several derivative rules to differentiate functions involving more than one operation on functions.
The basic derivative formulas in Section 11.4 allow you to differentiate a wide variety of functions. However, many of the economic functions we would like to analyze are products or quotients. Revenue is the product of the quantity sold and the unit price. Average cost is the quotient of the cost and the quantity produced. To find the derivative of either of these functions, we’ll need rules for finding the derivative of products or quotients.
Derivatives are perfect for examining change. By their definition, they tells us how one variable changes when another variable changes. In business and economics, this allows us to examine how revenue and cost change as the quantity produced and sold changes. Marginal revenue and marginal cost help a business determine compute these changes.
Elasticity is used to determine how changes in price affect the quantity demanded by consumers. Understanding this relationship helps us to determine whether a price should be increased or decreased.
In this section we’ll examine these terms and apply them to several examples drawn from businesses operating in the real world.
In section 11.3, we learned how to find the derivative of a function at a specific point. In this section, we define and show how to find the derivative function f ′(x). Using this function, we can find the slope of the tangent line or the instantaneous rate of change at any x value.
