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Basic Mathematics for Economists - Rosser - Chapter 12

12 Further topics in calculus Use the chain, product and quotient rules for differentiation. Choose the most appropriate method for differentiating different forms of functions. Check the second-order conditions for optimization of relevant economic functions using the quotient rule for differentiation. | 12 Further topics in calculus Learning objectives After completing this chapter students should be able to Use the chain product and quotient rules for differentiation. Choose the most appropriate method for differentiating different forms of functions. Check the second-order conditions for optimization of relevant economic functions using the quotient rule for differentiation. Integrate simple functions. Use integration to determine total cost and total revenue from marginal cost and marginal revenue functions. Understand how a definite integral relates to the area under a function and apply this concept to calculate consumer surplus. 12.1 Overview In this chapter some techniques are introduced that can be used to differentiate functions that are rather more complex than those encountered in Chapters 8 9 10 and 11. These are the chain rule the product rule and the quotient rule. As you will see in the worked examples it is often necessary to combine several of these methods to differentiate some functions. The concept of integration is also introduced. 12.2 The chain rule The chain rule is used to differentiate functions within functions . For example if we have the function y f z and we also know that there is a second functional relationship z g x then we can write y as a function of x in the form y f g x 1993 2003 Mike Rosser To differentiate y with respect to x in this type of function we use the chain rule which states that dy dy dz dx dz dx One economics example of a function within a function occurs in the marginal revenue productivity theory of the demand for labour where a firm s total revenue depends on output which in turn depends on the amount of labour employed. An applied example is explained later. However we shall first look at what is perhaps the most frequent use of the chain rule which is to break down an awkward function artificially into two components in order to allow differentiation via the chain rule. Assume for example that you wish to .