Question 1

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

Accepted Answer

$$3e^{3x}$$

Question 2

Suppose the price demand equation for a product is $$p = 50 - 0.2x$$, where $$x is $$the number of units sold. What does $$\frac{dp}{dx}$$ represent in this context?

Accepted Answer

The rate at which price changes with respect to the number of units sold

Question 3

What is the derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 4

If $$A(t) = 10,000e$$^{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 5

Which of the following statements about the graphs of $$y = b^x$$ and $$y = \log_b x is $$correct?

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

A sandwich shop franchise grows according to $$N(t) = 250 + 120\ln(t+1)$$, where $$t is $$years since 1980. Estimate the rate of change in the number of locations in 2028.

Accepted Answer

$$10$$ locations per year

Question 8

If $$f(x) = a^x$$ for $$a \u003e 0$$ and $$a 
eq 1$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$a^x \ln(a)$$

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

$$800$$

Question 11

Find the derivative of $$f(x) = \ln(5x).$$

Accepted Answer

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

Question 12

If $$f(x) = \log_b x$$, what is $$f'(x) in $$terms of $$b$$?

Accepted Answer

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

Question 13

Which of the following best describes why the power rule cannot be used to differentiate $$e^x$$?

Accepted Answer

Because $$e^x is $$not a power function of $$x$$

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

$$2x e$$^{x^2}$$

Derivatives of Exponential and Logarithmic Functions- Chapter 3

Study Guide - Practice Questions

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