e-xcos(nx)ln(2)⋅(-e-xcos(nx)+e-x(-sin(nx)⋅n))

Rewrite as ln(e-x⋅cos(nx))ln(2)

Question 1

Given y=log2(e-x&#x22C5;cos(nx)), which of the following is the correct expression for dydx?

Accepted Answer

e-xcos(nx)ln(2)&#x22C5;(-e-xcos(nx)+e-x(-sin(nx)&#x22C5;n))

Question 2

For y=lnxx2, what is the correct expression for d2ydx2 (the second derivative)?

Accepted Answer

x2(-1x-2lnx)-(xlnx-2x)&#x22C5;4x2x4

Question 3

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

Accepted Answer

cotxx&#x22C5;(sinx)lnx

Question 4

Which of the following is the correct simplification for ln(e-x&#x22C5;cos2xx2+x+1)?

Accepted Answer

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

Question 5

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

Accepted Answer

-1

Question 6

Given y=(sinx)lnx, which of the following best describes the domain of y?

Accepted Answer

x\u003e0 and sinx\u003e0

Question 7

Suppose y=log2(e-x&#x22C5;cos(nx)). For which values of x is y undefined?

Accepted Answer

When cos(nx)=0

Question 8

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

Accepted Answer

y approaches 0

Question 9

If y=log2(e-x&#x22C5;cos(nx)), what is the value of dydx at x=0?

Accepted Answer

-nln(2)

Question 10

Which of the following is the correct derivative of y=ln(e-x&#x22C5;cos2xx2+x+1)?

Accepted Answer

e-x&#x22C5;cos2xx2+x+1&#x22C5;(-2tanx-1)

Question 11

If y=(sinx)lnx, which of the following is a correct application of logarithmic differentiation?

Accepted Answer

lny=lnx&#x22C5;ln(sinx)

Question 12

Which of the following best describes the growth rate of y=lnxx2 as x increases?

Accepted Answer

It increases slowly at first, then decreases and approaches zero.

Question 13

If y=log2(e-x&#x22C5;cos(nx)), which of the following is the correct first step to find dydx?

Accepted Answer

Rewrite as ln(e-x&#x22C5;cos(nx))ln(2)

Question 14

For y=(sinx)lnx, which of the following is a necessary restriction on x for y to be real-valued?

Accepted Answer

x\u003e0 and sinx\u003e0

Advanced Differentiation Techniques in Calculus: Logarithmic and Trigonometric Functions

Study Guide - Practice Questions