Question 1

Which of the following is the correct derivative of the function $$f(x) = e$$^{3x}$$?

Accepted Answer

$$3e^{3x}$$

Question 2

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 3

Which of the following statements about the derivative of $$e^x is $$TRUE?

Accepted Answer

The derivative of $$e^x is$$ $$e^x.$$

Question 4

Given the price-demand equation $$p = 50 - 0.02x$$, where $$x is $$the number of items sold, what is the rate of change of price with respect to demand?

Accepted Answer

$$-0.02$$

Question 5

If $$A(t) = 10,000 e$$^{0.04t}$$ represents the amount in a continuously compounded account, what is the instantaneous rate of change of the amount after 2 years?

Accepted Answer

$$433.31$$

Question 6

Which of the following is the correct derivative of $$f(x) = \ln(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 7

Suppose the number of sandwich shop locations is modeled by $$N(t) = 250 + 120 \ln(t+1)$$, where $$t is $$years since 1980. What is the rate of change in the number of locations in 2028?

Accepted Answer

$$10$$

Question 8

Which of the following is the correct derivative of $$f(x) = b^x$$, where $$b \u003e 0$$ and $$b 
eq 1$$?

Accepted Answer

$$b^x \ln(b)$$

Question 9

If $$f(x) = \ln(5x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 10

A bank account grows according to $$A(t) = A_0$ e$$^{rt}$$. If the balance is 20$$,000$$ and the interest rate is 4$$\%$$, what is the instantaneous rate of change of the balance?

Accepted Answer

$$800$$

Question 11

Which of the following is NOT a property of the natural logarithm function $$\ln(x)$$?

Accepted Answer

Its domain is $$(-\infty, \infty).$$

Question 12

If $$f(x) = \ln(x^3$)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{3}{x}$$

Question 13

Which of the following best describes the relationship between the graphs of $$y = b^x$$ and $$y = \log_b x$$?

Accepted Answer

They are symmetric with respect to the line $$y = x.$$

Question 14

If $$f(x) = \ln(x^2$ + 4)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 4}$$

Question 15

Which of the following is the correct derivative of $$f(x) = \log_b x$$ for $$b \u003e 0$$, $$b 
eq 1$$?

Accepted Answer

$$\frac{1}{x \ln(b)}$$

Question 16

A company’s revenue is modeled by $$R(x) = 1000 e$$^{0.01x}$$, where x$ is the number of months. What is the rate of change of revenue when x = 12$?

Accepted Answer

Question 17

If $$f(x) = e$$^{2x+1}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x+1}$$

Question 18

Which of the following is the correct derivative of $$f(x) = \ln(ax)$$, where $$a \u003e 0$$?

Accepted Answer

$$\frac{1}{x}$$

Question 19

If $$f(x) = 5e$$^{0.03x}$$, what is f$$'(x)$$?

Accepted Answer

$$0.15e^{0.03x}$$

Question 20

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ - 1)$$?

Accepted Answer

$$\frac{2x}{x^2$ - 1}$$

Derivatives of Exponential and Logarithmic Functions: Business Calculus Study Guide

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