Question 1

Which of the following best describes a composite function?

Accepted Answer

A function formed by applying one function to the result of another function, such as $m(x) = f(g(x))$.

Question 2

Given $f(x) = x^2 + 1$ and $g(x) = 3x - 2$, what is $f(g(x))$?

Accepted Answer

$(3x - 2)^2 + 1$

Question 3

Which of the following is the correct statement of the Chain Rule for differentiation?

Accepted Answer

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

Question 4

If $y = (2x + 5)^4$, what is $\frac{dy}{dx}$ using the General Power Rule?

Accepted Answer

$8(2x + 5)^3$

Question 5

Suppose $h(x) = \sqrt{5x^2 + 3}$. What is $h'(x)$?

Accepted Answer

$\frac{1}{2\sqrt{5x^2 + 3}} \cdot 10x$

Question 6

Which of the following is a correct application of the Chain Rule to $y = \sin(3x)$?

Accepted Answer

$3\cos(3x)$

Question 7

A pharmaceutical company models the concentration of a drug in the bloodstream as $C(t) = e^{-0.2t^2}$. What is the rate of change of concentration with respect to time?

Accepted Answer

$-0.4t e^{-0.2t^2}$

Question 8

If $y = (x^3 + 2x)^5$, what is $\frac{dy}{dx}$?

Accepted Answer

$5(x^3 + 2x)^4 \cdot (3x^2 + 2)$

Question 9

Which of the following is the derivative of $f(x) = \ln(4x^2 + 1)$?

Accepted Answer

$\frac{8x}{4x^2 + 1}$

Question 10

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

Accepted Answer

$15(5x + 2)^2$

Question 11

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

Accepted Answer

$-4x \sin(2x^2)$

Question 12

Which of the following best illustrates the use of the Chain Rule in a real-world context?

Accepted Answer

Calculating the rate at which the area of a circular oil spill increases as the radius expands over time.

Question 13

Given $y = 	an(5x)$, what is $\frac{dy}{dx}$?

Accepted Answer

$5 \sec^2(5x)$

Question 14

If $y = e^{3x^2}$, what is $\frac{dy}{dx}$?

Accepted Answer

$6x e^{3x^2}$

Question 15

A company’s cost function is $C(q) = \sqrt{2q + 9}$. What is the marginal cost function?

Question 16

If $y = (x^2 + 1)^7$, what is $\frac{dy}{dx}$?

Accepted Answer

$14x(x^2 + 1)^6$

The Chain Rule and Composite Functions-Chapter 3

Study Guide - Practice Questions

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