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 $$C'(x)$$?

Accepted Answer

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

Guida di studio - Domande di pratica

Study guide - Flashcard