Question 1

Which of the following correctly states the product rule for differentiating the product of two functions $$f(x)$$ and $$g(x)$$?

Accepted Answer

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

Question 2

A company’s cost function is $$C(x) = (2x + 3)(4x - 1). $$What is $$C'(x)$$ using the product rule?

Accepted Answer

$$C'(x) = 2(4x - 1) + (2x + 3)4$$

Question 3

Given $$f(x) = x^5$ \cdot x^6$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 11x$$^{10}$$

Question 4

If $$y = \frac{3x^2$ + 2x}{x^2$ + 1}$$, what is $$y'$$ using the quotient rule?

Accepted Answer

$$y' = \frac{(6x + 2)(x^2$ + 1) - (3x^2$ + 2x)(2x)}{(x^2$ + 1)^2$}$$

Question 5

A business’s average revenue per unit is $$A(x) = \frac{R(x)}{x}. $$According to the quotient rule, what is $$A'(x)$$?

Accepted Answer

$$A'(x) = \frac{R'(x)x - R(x)}{x^2$}$$

Question 6

Differentiate $$y = (5x - 1)(3x^2$ - 7x + 10)$$ using the product rule.

Accepted Answer

$$y' = (5x - 1)(6x - 7) + (3x^2$ - 7x + 10) \cdot 5$$

Question 7

For $$f(x) = \frac{\sqrt{x} + 1}{2\sqrt{x} - 1}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 8

A pharmaceutical company models the concentration of a drug in the bloodstream as $$F(t) = \frac{1}{t - 4}. $$What is $$F'(t)$$?

Accepted Answer

$$F'(t) = -\frac{1}{(t - 4)^2$}$$

Question 9

If $$y = \frac{2x^2$ + 1}{5x^3$}$$, what is $$y'$$ using the quotient rule?

Accepted Answer

$$y' = \frac{(4x)(5x^3$) - (2x^2$ + 1)(15x^2$)}{(5x^3$)^2$$}$

Question 10

A company’s profit function is $$P(x) = (x^4$ + 1)(x^2$ - 1). $$What is $$P'(x)$$ using the product rule?

Accepted Answer

$$P'(x) = (x^4$ + 1) \cdot 2x + (x^2$ - 1) \cdot 4x^3$

Question 11

For $$F(x) = (-2\sqrt{x} - \frac{1}{4\sqrt{x}})(2 - 2x^2$)$$, what is $$F'(x)$$ using the product rule?

Accepted Answer

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

Question 12

A company’s revenue function is $$R(x) = (3\sqrt{x} + 2x)(x - 2\sqrt{x} - 1). $$What is $$R'(x)$$ using the product rule?

Accepted Answer

$$R'(x) = (3\sqrt{x} + 2x)'(x - 2\sqrt{x} - 1) + (3\sqrt{x} + 2x)(1 - \frac{2}{2\sqrt{x}})$$

Question 13

For $$f(x) = \frac{3\sqrt{x} - 7}{\sqrt{x} + 3}$$, what is $$f'(x)$$ using the quotient rule?

Accepted Answer

$$f'(x) = \frac{\frac{3}{2\sqrt{x}}(\sqrt{x} + 3) - (3\sqrt{x} - 7) \cdot \frac{1}{2\sqrt{x}}}{(\sqrt{x} + 3)^2$}$$

Question 14

If $$G(x) = (1 - \frac{3}{x^2$})(-x^2$ - x - 1)$$, what is $$G'(x)$$ using the product rule?

Accepted Answer

$$G'(x) = (1 - \frac{3}{x^2$})'(-x^2$ - x - 1) + (1 - \frac{3}{x^2$})(-2x - 1)$$

Question 15

For $$F(t) = 2\left(\frac{3t^2$ - 1}{t}\right)$$, what is $$F'(t)$$ using the quotient rule?

Accepted Answer

$$F'(t) = 2 \cdot \frac{(6t)(t) - (3t^2$ - 1)(1)}{t^2$}$$

Question 16

If $$y = \frac{x^2$ + 1}{x^3$ - 1}$$, what is $$y'$$ using the quotient rule?

Accepted Answer

$$y' = \frac{(2x)(x^3$ - 1) - (x^2$ + 1)(3x^2$)}{(x^3$ - 1)^2$}$$

Question 17

A company’s cost function is $$C(x) = (3x - 2)(x + 4). $$What is $$C'(x)$$ using the product rule?

Accepted Answer

$$C'(x) = (3x - 2)'(x + 4) + (3x - 2)(1)$$

Product and Quotient Rules in Business Calculus: Lecture 6 Study Guide

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