Question 1

Given the function $y = x^3(x^2 - 4x)^3$, use the chain rule to find $\frac{dy}{dx}$. Which of the following is the correct derivative?

Accepted Answer

$\frac{dy}{dx} = 3x^2(x^2 - 4x)^3 + x^3 \cdot 3(x^2 - 4x)^2 \cdot (2x - 4)$

Question 2

For the function $y = x^3(x^2 - 4x)^3$, at which $x$-values does the curve have horizontal tangent lines?

Accepted Answer

$x = 0, x = 4, x = \frac{8}{3}$

Question 3

Suppose you are asked to graph $y = x^3(x^2 - 4x)^3$ and show the points where the curve has horizontal tangent lines. Which of the following ordered pairs correctly represents one of these points?

Accepted Answer

$(0, 0)$

Question 4

Given $y = 5u^2 \cdot 3u$ where $u = x^2 + 1$, find $\frac{dy}{dx}$ using the chain rule.

Accepted Answer

$\frac{dy}{dx} = 30u^2 x$

Question 5

Let $f(x) = \frac{x^2 + 1}{x - 1}$ and $g(x) = \sqrt{x}$. What is $f(g(4))$?

Accepted Answer

$f(g(4)) = \frac{5}{2}$

Question 6

Which of the following best describes the chain rule for differentiation?

Accepted Answer

The derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Question 7

A pharmaceutical company models the concentration of a drug in the bloodstream by $C(t) = (2t + 1)^4$. What is the rate of change of concentration with respect to time at $t = 2$?

Accepted Answer

$C'(2) = 8(2t + 1)^3$

Question 8

A business models its profit as $P(x) = x^2 e^{3x}$. What is the derivative $P'(x)$?

Accepted Answer

$P'(x) = 2x e^{3x} + 3x^2 e^{3x}$

Question 9

If $h(x) = \sin(x^2)$, what is $h'(x)$?

Accepted Answer

$h'(x) = 2x \cos(x^2)$

Question 10

A company’s revenue is modeled by $R(x) = \ln(5x^2 + 1)$. What is $R'(x)$?

Accepted Answer

$R'(x) = \frac{10x}{5x^2 + 1}$

Question 11

Suppose $y = (3x - 2)^5$. What is $\frac{dy}{dx}$?

Accepted Answer

$\frac{dy}{dx} = 15(3x - 2)^4$

Question 12

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

Accepted Answer

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

Question 13

A population grows according to $P(t) = 100e^{0.05t}$. What is the instantaneous rate of change of the population at $t = 10$?

Accepted Answer

$P'(10) = 5e^{0.5} 	imes 100$

Question 14

If $y = \sqrt{2x + 1}$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{dy}{dx} = \frac{2}{2\sqrt{2x + 1}}$

Question 15

A company’s cost function is $C(x) = 4x^3 - 2x^2 + 7x - 5$. What is the marginal cost at $x = 2$?

Accepted Answer

$C'(2) = 48 - 8 + 7$

Question 16

If $y = \ln(x^2 + 1)$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{dy}{dx} = \frac{2x}{x^2 + 1}$

Question 17

A company’s revenue is modeled by $R(x) = x^2 \ln(x)$. What is $R'(x)$?

Accepted Answer

$R'(x) = 2x \ln(x) + x$

Question 18

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

Accepted Answer

$\frac{dy}{dx} = 2e^{2x}$

Question 19

Suppose $y = (x^2 + 3x)^4$. What is $\frac{dy}{dx}$?

Accepted Answer

$\frac{dy}{dx} = 4(x^2 + 3x)^3 (2x + 3)$

Business Calculus Quiz Study Notes: Differentiation and Applications

Study Guide - Practice Questions

Study Guide - Flashcards