(-e-x⁢cos(nx)+e-x⁢-sin(nx)⁢n)e-x⁢cos(nx)⁢ln2

(x2⁢(1-2lnx+2)-x⁢lnx⁢4x3)x4

x\u003e0 and x≠kπ for integer k

Question 1

Given y=log2(e-x&#8290;cos(nx)), which of the following is the correct expression for dydx?

Accepted Answer

(-e-x&#8290;cos(nx)+e-x&#8290;-sin(nx)&#8290;n)e-x&#8290;cos(nx)&#8290;ln2

Question 2

For y=lnxx2, what is the correct expression for dy2dx2 (the second derivative)?

Accepted Answer

(x2&#8290;(1-2lnx+2)-x&#8290;lnx&#8290;4x3)x4

Question 3

Given y=(sinx)lnx, what is the correct derivative dydx?

Accepted Answer

(sinx)lnx&#8290;cotxx

Question 4

Suppose y=ln(e-cosxx2+x+1). Which of the following is the correct simplified form for dydx?

Accepted Answer

(2x+1x2+x+1-2tanx-1)

Question 5

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

Accepted Answer

Rewrite using logarithms, then use implicit differentiation and the chain rule.

Question 6

If y=lnxx2, what is the value of dydx at x=1?

Accepted Answer

0

Question 7

Which of the following is the correct logarithmic expansion for ln(e-cosxx2+x+1)?

Accepted Answer

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

Question 8

Which of the following is a correct application of the chain rule for differentiating y=log2(e-x&#8290;cos(nx))?

Accepted Answer

Differentiate the inside function, then divide by the original argument times ln2.

Question 9

If y=(sinx)lnx, what is the domain of y?

Accepted Answer

x\u003e0 and x&ne;k&#x3C0; for integer k

Question 10

Which of the following real-world scenarios could be modeled by a function of the form y=(sinx)lnx?

Accepted Answer

The amplitude of a periodic signal that grows logarithmically with time.

Question 11

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

Accepted Answer

y approaches 0.

Question 12

Which of the following is the correct first derivative of y=lnxx2?

Accepted Answer

x2&#8290;1x-lnx&#8290;2xx4

Question 13

Which of the following is the correct application of the product rule for y=(sinx)lnx after rewriting using logarithms?

Accepted Answer

The product rule is not needed; use the chain rule after logarithmic differentiation.

Question 14

If y=log2(e-x&#8290;cos(nx)), what is the value of dydx at x=0?

Accepted Answer

-1ln2

Question 15

Which of the following best describes the effect of the logarithmic term in y=(sinx)lnx as x increases?

Accepted Answer

It causes the amplitude of the function to increase slowly for large x.

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

Study Guide - Practice Questions