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?

Accepted Answer

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

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