Question 1

Let y = 	ext{log}_$$2(e^{-x}$$ 	ext{cos}(	ext{ln}x)). What is y'?

Accepted Answer

y' = rac{$$-e^{-x}$$ 	ext{cos}(	ext{ln}x) + $$e^{-x}(-	$$ext{sin}(	ext{ln}x)) rac{1}{x}}{$$e^{-x}$$ 	ext{cos}(	ext{ln}x) 	ext{ln}2}

Question 2

Given y = rac{	ext{ln}x}{$$x^2$$}, what is y''?

Accepted Answer

y'' = rac{$$x^2(1-2	$$ext{ln}x+2) - (x-	ext{ln}x 	imes 2x) 	imes 4x}{$$x^4$$}

Question 3

For y = (	ext{sin}$$x)^{	ext{ln}x$$}, what is y'?

Accepted Answer

y' = (	ext{sin}$$x)^{	ext{ln}x$$} rac{	ext{cot}x}{x}

Question 4

Given y = 	ext{ln}igg(rac{$$e^{-x}$$ 	ext{cos}^2 x}{$$x^2 + x + 1$$}igg), which of the following is the correct expression for y'?

Accepted Answer

y' = rac{-2x-1}{$$x^2 + x + 1$$} - 2 	ext{tan}x - 1

Question 5

Which of the following best describes the process used to differentiate y = (	ext{sin}$$x)^{	ext{ln}x$$}?

Accepted Answer

Logarithmic differentiation, using 	ext{ln}y = 	ext{ln}x 	ext{ln}(	ext{sin}x)

Question 6

Suppose a population of bacteria grows according to P(t) = $$e^{-t}$$ 	ext{cos}(	ext{ln}t). What is the rate of change of the population at time t?

Accepted Answer

P'(t) = $$-e^{-t}$$ 	ext{cos}(	ext{ln}t) + $$e^{-t}(-	$$ext{sin}(	ext{ln}t)) rac{1}{t}

Question 7

If y = rac{	ext{ln}x}{$$x^2$$}, which of the following best describes the behavior of y as x 	o 	ext{infty}?

Accepted Answer

y 	o 0

Question 8

Which of the following is the correct application of the chain rule for y = 	ext{log}_2(f(x))?

Accepted Answer

y' = rac{f'(x)}{f(x) 	ext{ln}2}

Question 9

A function is given by y = 	ext{ln}$$(e^{-x}$$ 	ext{cos}^2 x) - 	ext{ln}$$(x^2 + x + 1). $$Which of the following is equivalent to y?

Accepted Answer

y = -x + 2	ext{ln}(	ext{cos}x) - 	ext{ln}$$(x^2 + x + 1)$$

Question 10

Which of the following is the correct derivative of y = 	ext{ln}$$(e^{-x}$$ 	ext{cos}^2 x)?

Accepted Answer

y' = -1 - 2	ext{tan}x

Question 11

If y = 	ext{log}_$$2(e^{-x}$$ 	ext{cos}(	ext{ln}x)), which of the following is a correct first step in finding y'?

Accepted Answer

Rewrite as y = rac{	ext{ln}$$(e^{-x}$$ 	ext{cos}(	ext{ln}x))}{	ext{ln}2}

Question 12

Which of the following is the correct application of the product rule for y = x 	ext{ln}x?

Accepted Answer

y' = 1 	imes 	ext{ln}x + x 	imes rac{1}{x}

Question 13

Suppose y = rac{	ext{ln}x}{$$x^2$$} models the concentration of a drug in the bloodstream over time. What does the second derivative y'' represent in this context?

Accepted Answer

The rate at which the rate of change of drug concentration is changing

Question 14

If y = (	ext{sin}$$x)^{	ext{ln}x$$}, which of the following is a correct step in logarithmic differentiation?

Accepted Answer

ext{ln}y = 	ext{ln}x 	ext{ln}(	ext{sin}x)

Question 15

Which of the following is the correct derivative of y = 	ext{ln}$$(x^2 + x + 1$$)?

Accepted Answer

y' = rac{2x+1}{$$x^2 + x + 1$$}

Question 16

If y = 	ext{ln}$$(e^{-x}$$ 	ext{cos}^2 x) - 	ext{ln}$$(x^2 + x + 1$$), what is the value of y when x = 0?

Accepted Answer

y = 0

Advanced Differentiation Techniques: Logarithmic Differentiation and Derivatives of Exponential, Logarithmic, and Trigonometric Functions

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