Question 1

Which of the following correctly applies the Power Rule to find the derivative of $$f(x) = 12x^3$?

Accepted Answer

$$f'(x) = 36x^2$

Question 2

A medication is administered at a constant rate, modeled by $$g(x) = 8. $$What is the rate of change of the medication in the bloodstream?

Accepted Answer

$$g'(x) = 0$$

Question 3

Given $$y = \frac{2}{x^4$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$y' = -8x$$^{-5}$$

Question 4

Find the derivative of $$f(x) = x^5$ + \frac{4}{3}x^2$ - 5x + 1.$$

Accepted Answer

$$f'(x) = 5x^4$ + \frac{8}{3}x - 5$$

Question 5

A biologist models the growth of bacteria with $$h(x) = 6\sqrt{x^2$/3}. $$What is $$h'(x)$$?

Accepted Answer

$$h'(x) = 4x/\sqrt{3}$$

Question 6

Use the quotient rule to find the derivative of $$y = \frac{8x + 7}{9x - 5}.$$

Accepted Answer

$$y' = \frac{9(8x+7) - 8(9x-5)}{(9x-5)^2$}$$

Question 7

Given $$f(x) = (7 - x^2$)(x^3 - 5x + 1$$)$$, what is f$$'(x)$ using the product rule?

Accepted Answer

$$f'(x) = (7 - x^2$)(3x^2 - 5$$) + $$(x^3 - 5x + 1)(-2x$$)$

Question 8

Find the derivative of $$y = \frac{4x^3$}{3} - 7x + 8e^x.$$

Accepted Answer

$$y' = 4x^2$/3 - 7 + 8e^x$$

Question 9

A population grows according to $$y = 9x^9$e^x. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$y' = 9x^9$e^x + 81x^8$e^x$$

Question 10

Find the first and second derivatives of $$y = -8x^5$ - 7.$$

Accepted Answer

First: $$y' = -40x^4$, Second: y$$'' = $$-160x^3$$

Question 11

Given $$y = \frac{1}{2}x^4$ - \frac{5}{2}$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = 12x^2$

Question 12

Find the first derivative of $$h(x) = x^3$e^x$$ using the product rule.

Accepted Answer

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

Question 13

Given $$y = \frac{x^4$}{6} - \frac{5}{9}x^2$ - 6x$$, what is the third derivative $$y'''$$?

Accepted Answer

$$y''' = 4$$

Question 14

Does the curve $$y = x^4$ - 2x^2$ + 2$$ have any horizontal tangent lines? If so, where?

Accepted Answer

Yes, at $$x = 0$$ and $$x = \pm \sqrt{2}$$

Question 15

Which rule is most appropriate for differentiating $$f(x) = (7 - x^2$)(x^3 - 5x + 1$$)$?

Accepted Answer

Product Rule

Question 16

A chemical reaction rate is modeled by $$y = 8e^x. $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$8e^x$$

Question 17

Which of the following is the correct application of the sum rule for derivatives?

Accepted Answer

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

Differentiation Rules and Applications in Calculus I

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