Mathematics for Economists
Chapter 5 : Derivatives and limits
By Lund University
This chapter is entirely devoted to the derivative of a function of one variable. The derivative of function is defined as a limit of a specific ratio (the Newton quotient) and we begin the chapter with a brief introduction to limits. In the rest of the chapter we will learn how to differentiate various functions. To our help we will have a bunch of rules such as the chain rule. We will also need higher order derivatives. For example, we can sometimes use the second order derivative to distinguish between a maximum point and a minimum point. This chapter is concluded with a few more advanced topics such as implicit differentiation.
Limits and continuity
The main objective of this chapter is to study the derivative of a function. However, in order to understand the definition of a derivative we must look at limits. The limit of a function is the value that a function takes when x gets close to, but is not exactly equal to, a given value. Limits are closely related to another concept called continuity. Informally, a function is continuous if its graph is “connected”.
Rough introduction to limits
Rough introduction to limits at infinity
Rough introduction to infinite limits
Rough introduction to continuity
Limit laws
Continuity laws
Limits and continuity: Problems
Exercises on limits and continuity
Problem: Calculate limits
Problem: Calculate limits
Problem: Calculate limit
Problem: Calculate limit by trying a small value
Problem: Calculate limit by trying a small value
Problem: Calculate limit
Problem: Determine if a function is continuous
Problem: Determine if a function is continuous
Problem: Determine if a function is continuous
Problem: Make a function continuous
Basic derivatives
This section introduces derivatives. It begins by defining the tangent, a straight line that just touches the graph of the function. The slope of this tangent is precisely the derivative of the function at the touching point. From this, the formal definition of a derivative is presented as the limit of the Newton quotient. We then look at rules which will help us finding the derivative of a function. Finally we look at the relationship between derivatives and whether the function is increasing or decreasing.
Introduction to derivatives
Derivatives
Rules for differentiation
Derivatives and its relationship to increasing and decreasing functions
Basic derivatives: Problems
Exercises on derivatives
Problem: Find derivatives from the graph
Problem: Find the derivative using the definition
Problem: Find the derivative using the definition and the equation of the tangent
Problem: Find the slope of the tangent
Problem: Economic interpretation of slope and intercept
Problem: Find marginal profit, revenue and cost
Problem: Compute derivatives
Problem: Find the derivative of a composite function
Problem: Compute derivatives
Problem: Differentiate functions
Problem: Differentiate functions
Problem: Differentiate functions
Problem: Differentiate functions
Problem: Determine when a function is increasing/decreasing
Problem: Determine when a function is increasing
Problem: Economic interpretation of the derivative of an exponential function
Problem: Find the equation of the tangent
Chain rule
This section is devoted to the chain rule. More complex functions can be written as a composition of simpler functions. Such functions can be differentiated using the chain rule where we only need to differentiate the simpler functions.
Chain rule
Problem: Differentiate using the chain rule
Problem: Differentiate function
Problem: Compute derivatives
Problem: Compute derivatives
Problem: Determine when a function is increasing
Problem: Find derivatives
Higher order derivatives
By differentiating the derivative of a function we get what is called the second derivative. The same idea can be extended to higher order derivatives. Second derivatives will be important in the next chapter. In this section we also look at the relationship between second derivatives and whether the function is convex or concave.
Higher order derivatives
Derivatives and its relationship to convex and concave functions
Problem: Compute second derivatives
Problem: Compute second derivative
Problem: Compute higher order derivatives
Problem: Find first and second order derivatives
Problem: Find first and second order derivatives
Implicit differentiation and the derivative of the inverse
This section contains two topics that are a bit more advanced. First, we look at implicit differentiation. This is a method that allows us to find a derivative when we only have an implicit relationship between two variables. Second, we look at a method which allows us to find the derivative of the inverse of the function without actually finding the inverse function.
Implicit differentiation
Inverse function theorem
Derivative of the inverse of a function
Implicit differentiation and the derivative of the inverse: Problems
Exercises on implicit differentiation and the derivative of the inverse