## 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