# 5 The Derivative

## 5.1 The Derivative

### Practice Problems

**Problem 5.1.1: **Find the derivative for the following functions. Additionally, find the slope of the function when x=2.

a)

b)

c)

d)

e)

f)

g)

h)

i)

Answers

a)

b)

c)

d)

e)

f)

g)

h)

**Problem 5.1.2: **For what value(s) of x does the function below have a horizontal tangent line?

Answer: x=-2, 3

**Problem 5.1.3: **Determine the coefficients of a and b that make the statement below true:

Answer: a=5, b=-6

**Problem 5.1.4: **A company finds that its revenue function is

Calculate the marginal revenue for the 10th unit.

Answer: 1,840

**Problem 5.1.5: **The demand function for a product is given below:

a) Calculate the marginal revenue when the firm sells its 20th unit.

b) At what level of production does marginal revenue equal 0?

Answer: 460, 250

**Problem 5.1.6: **Consider the function h(x) for which h(3)=6 and h'(3)=-5. Find f'(3) for the function

Answer: -7/3

**Problem 5.1.7: **Find the derivative for the following functions. Additionally, find the slope of the function when x=2.

a)

b)

c)

d)

Answers

a)

b)

c)

d)

### External Resources

Khan Academy: Power Rule

Khan Academy: Differentiating Integer Powers

Khan Academy: Product Rule (Ignore the Trig)

Khan Academy: Quotient Rule (Ignore the Trig)

Khan Academy: Chain Rule

Khan Academy: Worked Chain Rule Example

## 5.2 Exponentials and logarithms: A Review

### Practice Problems

**Problem 5.2.1: **You place 1,000USD in an account which pays 6% compounded continuously. Find the future value of the account after 5 years? 10 years? 20 years? What if the account paid 8%?

Answers: 1,350, 1,822, 3,320; 1,492, 2,226, 4,953

**Problem 5.2.2: **After a child’s birth, an account is opened, and a single payment has been made, so what when the child is 18, they will receive 25,000USD. What initial deposit was made assuming the account earned 4% per year compounded continuously.

Answer: 12,169

**Problem 5.2.3: **The real GDP per capita of a country in a given year is 3,500USD. How long will it take for the GDP per capita to reach 10,000USD if the economy grows at a rate of 2% compounded continuously.

Answer: 52.5 years

**Problem 5.2.4: **Mimi deposits 10,000USD into an account which compounds continuously. After 30 years, the deposit is worth 50,000USD. What interest rate did this account pay?

Answer: 5.4%

**Problem 5.2.5: **Fully expand the following logarithms:

a)

b)

Answers:

a)

b)

**Problem 5.2.6: **Solve the following for x:

a)

b)

c)

d)

Answers:

a)

b)

c)

d)

### External Resources

Khan Academy: e and Compound Interest

Khan Academy: Evaluating ln with a Calculator

## 5.3 The Derivative of implicit Equations

### Practice Problems

**Problem 5.3.1: **Find dy/dx for the following:

a)

b)

c)

d)

Answers:

a)

b)

c)

d)

### External Resources

Khan Academy: Implicit Differentiation

## 5.4 The Derivatives of exponential and logarithmic functions

### Practice Problems

**Problem 5.4.1:** Find the derivative of the following functions:

a)

b)

c)

d)

Answers:

a)

b)

c)

d)

**Problem 5.4.2:** Find the derivative of the following functions:

a)

b)

c)

d)

e)

Answers:

a)

b)

c)

d)

e)

**Problem 5.4.3:** Find the derivative of the following:

Answer:

### External Resources

Khan Academy: Derivative of a Natural Log

Khan Academy: Worked Example