Question 1

Given the function $$h(x) = (x^3$ - 3)(x^4$ + 8)$$, what is $$h'(x)$$?

Accepted Answer

$$h'(x) = 3x^2$(x^4 + 8$$) + $$4x^3(x^3$ - 3)$$

Question 2

Which of the following correctly applies the Quotient Rule to $$f(x) = \dfrac{x^2$ - 4}{x + 2}$$?

Accepted Answer

$$f'(x) = \dfrac{(x + 2)2x - (x^2$ - 4) \cdot 1}{(x + 2)^2$}$$

Question 3

Suppose $$f(x) = x^3$ - x^2$ - 12x + 1. $$For which $$x$$ does $$f(x)$$ have a horizontal tangent line?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 4

If $$f(x) = \dfrac{1}{\sqrt{x}}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -\dfrac{1}{2x$$^{3/2}$$}$$

Question 5

Given $$g(x) = x^3$ f(x)$$ and $$f(2) = 2$$, $$f'(2) = 3$$, what is $$g'(2)$$?

Accepted Answer

$$g'(2) = 3 \cdot 4 \cdot 2 + 8 \cdot 3 = 24 + 24 = 48$$

Question 6

Which of the following is the correct derivative of $$y = \dfrac{5}{14x^7$}$$?

Accepted Answer

$$y' = -\dfrac{5}{14} \cdot 7x$$^{-8}$$

Question 7

If $$h(x) = \dfrac{f(x)}{x-3}$$, and $$f(2) = 2$$, $$f'(2) = 3$$, what is $$h'(2)$$?

Accepted Answer

$$h'(2) = \dfrac{f'(2)(2-3) - f(2) \cdot 1}{(2-3)^2$} = \dfrac{3(-1) - 2}{1} = -3 - 2 = -5$$

Question 8

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

Accepted Answer

$$f'(x) = 0$$

Question 9

Let $$f(x) = x^n$$ where $$n is a $$rational number. What is $$f'(x)$$?

Accepted Answer

$$f'(x) = n x$$^{n-1}$$

Question 10

Given $$y = \dfrac{5}{(2x)^4$}$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{dy}{dx} = -\dfrac{5}{8x^5$}$$

Question 11

If $$f(x) = 3x^2$ - 7x$$^{1/2}$$, what is f$$''(x)$$?

Accepted Answer

$$f''(x) = 6 - \dfrac{7}{4}x$$^{-3/2}$$

Question 12

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

Accepted Answer

$$f'(a) = 0$$

Question 13

Given $$h(t) = \dfrac{1}{t$$^{1/3}$$}$$, what is $$h'(t)$$?

Accepted Answer

$$h'(t) = -\dfrac{1}{3} t$$^{-4/3}$$

Question 14

If $$f(x) = \sqrt[3]{x}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{1}{3} x$$^{-2/3}$$

Question 15

Which of the following is the correct application of the Product Rule for $$\dfrac{d}{dx}[f(x)g(x)]$$?

Accepted Answer

$$f'(x)g(x) + f(x)g'(x)$$

Question 16

Suppose $$f(x) = x^2$ - 4$$ and $$g(x) = x + 2. $$Using the Quotient Rule, what is $$\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)$$?

Accepted Answer

$$\dfrac{(x + 2) \cdot 2x - (x^2$ - 4) \cdot 1}{(x + 2)^2$}$$

Question 17

If $$f(x) = 6x$$^{1/2}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 3x$$^{-1/2}$$

Question 18

A real-world application: The position of a car is given by $$s(t) = 3t^2$ - 7t$$^{1/2}$$. What is the car's acceleration at t = 4$ seconds?

Accepted Answer

$$a(4) = 6 - \dfrac{7}{4} \cdot 4$$^{-3/2}$$ = 6 - \dfrac{7}{4} \cdot \dfrac{1}{8} = 6 - \dfrac{7}{32} = \dfrac{185}{32}$$

Question 19

If $$f(x) = x^n$$ and $$n is a $$rational number, what is the $$kth $$derivative $$f^{(k)}(x$$)$$?

Accepted Answer

$$f^{(k)}(x) = n(n-1$$)\$$cdots(n-k+1)x^{n-k}$$

Derivatives: Product Rule, Quotient Rule, and Power Rule

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