Question 1

Which of the following correctly states the Product Rule for the derivative of two differentiable functions $f(x)$ and $g(x)$?

Accepted Answer

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

Question 2

Suppose $h(x) = (2x^2 + 3x)(x^3 - 4)$. What is $h'(x)$ using the Product Rule?

Accepted Answer

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

Question 3

Given $f(x) = x^2$ and $g(x) = e^x$, what is the derivative of $f(x)g(x)$?

Accepted Answer

$ 2x e^x + x^2 e^x $

Question 4

Which of the following is the correct Quotient Rule for differentiable functions $T(x)$ (top) and $B(x)$ (bottom)?

Accepted Answer

$ \frac{T'(x)B(x) - T(x)B'(x)}{[B(x)]^2} $

Question 5

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

Accepted Answer

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

Question 6

A company’s revenue is given by $R(x) = (5x + 2)(x^2 - 3x)$. What is the marginal revenue, $R'(x)$?

Accepted Answer

$ R'(x) = (5x + 2) \cdot (2x - 3) + (x^2 - 3x) \cdot 5 $

Question 7

Let $f(x) = x^2 + 1$ and $g(x) = x - 2$. At what value of $x$ is the tangent line to $h(x) = f(x)g(x)$ horizontal?

Accepted Answer

$ x = 2 $

Question 8

Suppose the total sales $S$ (in thousands) of a product $t$ months after launch is $S(t) = \frac{100,000t}{t + 6}$. What does $S'(10)$ represent in this context?

Accepted Answer

The rate at which sales are increasing after 10 months, in thousands of games per month.

Question 9

Given $f(x) = x^3$ and $g(x) = \sin(x)$, what is the derivative of $f(x)g(x)$?

Accepted Answer

$ 3x^2 \sin(x) + x^3 \cos(x) $

Question 10

If $y = \frac{e^x}{x^2}$, what is $y'$?

Accepted Answer

$ y' = \frac{e^x x^2 - e^x \cdot 2x}{x^4} $

Question 11

A company’s cost function is $C(x) = (x^2 + 4x)(2x - 5)$. What is the marginal cost $C'(x)$?

Question 12

If $y = \frac{x^2 + 1}{x}$, what is $y'$?

Accepted Answer

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

Question 13

A company’s profit function is $P(x) = (x^2 - 5x)(3x + 2)$. What is the marginal profit $P'(x)$?

Accepted Answer

$ P'(x) = (x^2 - 5x) \cdot 3 + (3x + 2) \cdot (2x - 5) $

Question 14

If $f(x) = x^2$ and $g(x) = x + 1$, what is the derivative of $\frac{f(x)}{g(x)}$?

Accepted Answer

$ \frac{2x(x + 1) - x^2(1)}{(x + 1)^2} $

Question 15

A company’s demand function is $D(p) = (100 - p)(p^2 + 1)$. What is the marginal demand $D'(p)$?

Accepted Answer

$ D'(p) = (100 - p) \cdot 2p + (p^2 + 1) \cdot (-1) $

Question 16

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

Accepted Answer

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

Question 17

A company’s revenue is $R(x) = (x^2 + 2x)(x - 3)$. If the company sells 5 units, what is the rate of change of revenue at $x = 5$?

Accepted Answer

$ R'(5) = (2 \cdot 5 + 2)(5 - 3) + (5^2 + 2 \cdot 5)(1) $

Derivatives of Products and Quotients- Chapter 3

Study Guide - Practice Questions

Study Guide - Flashcards