Question 1

Which of the following correctly applies the Power Rule to find the derivative of $$f(x) = x^5$?

Accepted Answer

$$f'(x) = 5x^4$

Question 2

Given $$y = \frac{1}{x^3$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = -3x$$^{-4}$$

Question 3

If $$f(x) = 7x^2$ - 4x + 5$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 14x - 4$$

Question 4

A medication's concentration in the bloodstream is modeled by $$C(t) = 5t$$^{1/2}$$, where t$ is time in hours. What is the rate of change of concentration at t = 4$ hours?

Accepted Answer

$$\frac{dC}{dt} = 1.25$$

Question 5

Which of the following statements about the derivative of a constant function $$f(x) = c is $$true?

Accepted Answer

The derivative is always zero.

Question 6

If $$f(x) = 3x^2$ and g(x) = 2x$, what is $$\frac{d}{dx}[f(x) + g(x)]$$?

Accepted Answer

$$6x + 2$$

Question 7

A population of bacteria is modeled by $$P(t) = 100e$$^{0.02t}$$. What is the instantaneous rate of change of the population at t = 0$?

Accepted Answer

$$2$$

Question 8

Which of the following is the correct derivative of $$y = \sqrt{x}$$?

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 9

If $$f(x) = 2x - \frac{1}{2}e^x$$, what is the slope of the tangent line at $$x = 0$$?

Accepted Answer

$$\frac{3}{2}$$

Question 10

Which of the following is the correct application of the Constant Multiple Rule for $$y = 5x^4$?

Accepted Answer

$$y' = 20x^3$

Question 11

Given $$f(x) = x^3$ - 2x^2$ + x - 5$$, what is $$f''(x)$$, the second derivative?

Accepted Answer

$$f''(x) = 6x - 4$$

Question 12

A function $$f(x)$$ has a horizontal tangent at $$x = a if$$:

Accepted Answer

$$f'(a) = 0$$

Question 13

If $$y = \frac{7}{3x$$^{-2}$$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$14x$$

Question 14

Which of the following is the correct derivative of $$y = 3\pi x$$?

Accepted Answer

$$3\pi$$

Question 15

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

Accepted Answer

$$e^x$$

Question 16

A company’s profit is modeled by $$P(x) = 4x^3$ - 6x^2$ + 2x + 10. $$What is the rate of change of profit when $$x = 2$$?

Accepted Answer

$$P'(2) = 20$$

Question 17

Which of the following is the correct third derivative of $$f(x) = x^4$?

Accepted Answer

$$f'''(x) = 24x$$

Question 18

If $$y = \frac{1}{2\sqrt[3]{x^2$}}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$-\frac{1}{3x$$^{5/3}$$}$$

Question 19

Which of the following is the correct application of the Sum Rule for $$h(x) = x^2$ + 3x + 4$$?

Accepted Answer

$$h'(x) = 2x + 3$$

Question 20

If $$f(x) = 2x - \frac{1}{2}e^x$$, at what $$x-$$value is the tangent line to the graph of $$f$$ horizontal?

Accepted Answer

$$x = \ln 4$$

Rules of Differentiation: Power, Constant Multiple, Sum, and Higher-Order Derivatives

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