Q: Let y=log2(e-x⁢cos(nx)). What is dydx?

e-x⁢cos(nx)⁢(-ntan(nx)-1)e-x⁢cos(nx)⁢ln2

Q: If y=log2(e-x⁢cos(nx)), which of the following is a correct first step in finding dydx?

Rewrite the logarithm in terms of natural logarithm: ln(e-x⁢cos(nx))ln2

Question 1

Let y=log2(e-x&#x2062;cos(nx)). What is dydx?

Accepted Answer

e-x&#x2062;cos(nx)&#x2062;(-ntan(nx)-1)e-x&#x2062;cos(nx)&#x2062;ln2

Question 2

Given y=lnxx2, what is dydx?

Accepted Answer

1-2xlnxx3

Question 3

For y=lnxx2, what is the second derivative y''?

Accepted Answer

x(1-2xlnx)-(x-2xlnx)&#x2062;4x2x6

Question 4

Let y=(sinx)lnx. What is dydx?

Accepted Answer

(sinx)lnx&#x2062;cotx&#x2062;lnx+1x

Question 5

Given y=ln(e-x&#x2062;cos2xx2+x+1), what is dydx?

Accepted Answer

2x+1x2+x+1-2tanx-1

Question 6

Which of the following best describes the process used to differentiate y=(sinx)lnx?

Accepted Answer

Logarithmic differentiation is used, taking the natural logarithm of both sides before differentiating.

Question 7

If y=log2(e-x&#x2062;cos(nx)), which of the following is a correct first step in finding dydx?

Accepted Answer

Rewrite the logarithm in terms of natural logarithm: ln(e-x&#x2062;cos(nx))ln2

Question 8

Suppose y=ln(e-x&#x2062;cos2xx2+x+1). Which of the following is the correct simplification for y before differentiating?

Accepted Answer

-x+2ln(cosx)-ln(x2+x+1)

Question 9

Which of the following is the correct application of the chain rule for y=ln(e-x&#x2062;cos2x)?

Accepted Answer

y'=-1-2tanx

Question 10

A biologist models the growth of a population with y=ln(e-x&#x2062;cos2x/(x2+x+1)). What does the term -x in the simplified logarithmic expression represent in terms of population change?

Accepted Answer

A constant rate of exponential decay in the population.

Question 11

Which of the following is the correct derivative of y=(sinx)lnx using logarithmic differentiation?

Accepted Answer

(sinx)lnx&#x2062;cotx&#x2062;lnx+1x

Question 12

If y=lnxx2, which of the following best describes the behavior of y as x approaches infinity?

Accepted Answer

y approaches 0.

Question 13

Which of the following is a correct application of the quotient rule for y=lnxx2?

Accepted Answer

x2&#x2062;1x-lnx&#x2062;2xx4

Question 14

A medical researcher models the concentration of a drug in the bloodstream as y=log2(e-x&#x2062;cos(nx)). What does the cos(nx) term represent in this context?

Accepted Answer

A periodic fluctuation in drug concentration due to dosing intervals.

Question 15

Which of the following is the correct simplification for ln(e-x&#x2062;cos2x/(x2+x+1))?

Accepted Answer

-x+2ln(cosx)-ln(x2+x+1)

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

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