Question 1

Which of the following is the correct derivative of the function $$f(x) = 7x^4$ - 3x + 5$$?

Accepted Answer

$$f'(x) = 28x^3$ - 3$$

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = 5e^t - 2t^2$. What is the rate of change of the drug concentration at any time t$?

Accepted Answer

$$C'(t) = 5e^t - 4t$$

Question 3

Which of the following statements about the derivative of a product of two functions is correct?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$$

Question 4

Find the derivative of $$h(x) = \frac{9 + 8x - 6x}{x}.$$

Accepted Answer

$$h'(x) = -\frac{9}{x^2$}$$

Question 5

A population of bacteria grows according to $$P(t) = 2e^t + 3t^2$. What is the equation of the tangent line to P(t$$)$$ at t = 0$?

Accepted Answer

$$y = 2 + 3t$$

Question 6

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{x^2$ e^x}{x^3$ + 5e^x}$$?

Accepted Answer

$$f'(x) = \frac{(x^3$ + 5e^x)[2x e^x + x^2$ e^x] - x^2$ e^x [3x^2$ + 5e^x]}{(x^3$ + 5e^x)^2$}$$

Question 7

For which value(s) of $$x is $$the tangent line to $$y = x^3$ - 3x - 2$$ horizontal?

Accepted Answer

$$x = 1, x = -1$$

Question 8

Given $$y = (5x - 1)(2x^2$ + x)$$, what is the slope of the tangent line at $$x = 1$$?

Accepted Answer

$$27$$

Question 9

Which of the following is the correct second derivative of $$f(x) = 6e^x - 2x^2$ + 3$$?

Accepted Answer

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

Question 10

A chemical reaction's rate is modeled by $$R(t) = 18e^t - \frac{1}{t}. $$What is $$R'(t)$$?

Accepted Answer

$$R'(t) = 18e^t + \frac{1}{t^2$}$$

Question 11

Which of the following is NOT a correct differentiation rule?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g'(x)$$

Question 12

Find the derivative of $$g(x) = \frac{1}{x} + 4x.$$

Accepted Answer

$$g'(x) = -\frac{1}{x^2$} + 4$$

Question 13

A car's position is given by $$s(t) = t^3$ - 3t - 2. $$What is the car's acceleration at $$t = 2$$?

Accepted Answer

$$a(2) = 12$$

Question 14

Which of the following is the correct derivative of $$r(t) = \frac{2}{t^2$} + \frac{8}{t^4$}$$?

Accepted Answer

$$r'(t) = -\frac{4}{t^3$} - \frac{32}{t^5$}$$

Question 15

Given $$y = \frac{6e^x}{1 + e^x}$$, what is $$y'(x)$$?

Accepted Answer

$$y'(x) = \frac{6e^x(1 + e^x) - 6e^x e^x}{(1 + e^x)^2$}$$

Question 16

Which of the following best describes the meaning of the second derivative $$f''(x) in a $$real-world context?

Accepted Answer

It represents the rate of change of the rate of change of $$f(x).$$

Question 17

If $$f(x) = e^x + 4x^2$ + 3$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = e^x + 8x$$

Question 18

Which of the following is the correct equation of the tangent line to $$y = 1 + 6e^x at $$the point $$(0, 7)$$?

Accepted Answer

$$y = 7 + 6x$$

Differentiation Rules and Applications

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