# 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)

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?

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

Problem 5.1.4: A company finds that its revenue function is

Calculate the marginal revenue for the 10th unit.

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?

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

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)

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.

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.

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?

Problem 5.2.5: Fully expand the following logarithms:

a)
b)

a)
b)

Problem 5.2.6: Solve the following for x:

a)
b)
c)
d)

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)

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)

a)
b)
c)
d)

Problem 5.4.2: Find the derivative of the following functions:

a)
b)
c)
d)
e)

a)
b)
c)
d)
e)

Problem 5.4.3: Find the derivative of the following:

### External Resources

Khan Academy: Derivative of a Natural Log

Khan Academy: Worked Example