e-xcos(nx)⁢ln21⁢(-e-xcos(nx)+e-x(-sin(nx)⁢n))

x2(1-2lnx+2lnx)-x(x-lnx⁢2)⁢4x3x4

(2x+1x2+x+1-2tanx-1)⁢e-xcos2xx2+x+1

1e-xcos(nx)⁢ddx(e-xcos(nx))⁢1ln2

Question 1

Given y=log2(e-x&#x2062;cos(nx)), what is dydx?

Accepted Answer

e-xcos(nx)&#x2062;ln21&#x2062;(-e-xcos(nx)+e-x(-sin(nx)&#x2062;n))

Question 2

If y=lnxx2, what is dy2dx2?

Accepted Answer

x2(1-2lnx+2lnx)-x(x-lnx&#x2062;2)&#x2062;4x3x4

Question 3

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

Accepted Answer

cotxx&#x2062;(sinx)lnx

Question 4

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

Accepted Answer

(2x+1x2+x+1-2tanx-1)&#x2062;e-xcos2xx2+x+1

Question 5

Which of the following is the correct logarithmic differentiation step for y=(sinx)lnx?

Accepted Answer

lny=lnx&#x2062;ln(sinx)

Question 6

Suppose y=lnxx2. Which of the following best describes the behavior of y as x approaches infinity?

Accepted Answer

y approaches 0

Question 7

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

Accepted Answer

ddxlog2u=1u&#x2062;dudx&#x2062;1ln2

Question 8

Which of the following is the correct application of the product rule for y=x&#x2062;ln(cosx)?

Accepted Answer

y'=ln(cosx)+x&#x2062;-sinxcosx

Question 9

Which of the following best describes the use of logarithmic differentiation for the function y=(sinx)lnx?

Accepted Answer

It is useful because both the base and the exponent are functions of x.

Question 10

Given y=ln(e-xcos2xx2+x+1), which of the following is the correct simplification for y?

Accepted Answer

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

Question 11

Which of the following is the correct chain rule application for y=log2(e-x&#x2062;cos(nx))?

Accepted Answer

1e-xcos(nx)&#x2062;ddx(e-xcos(nx))&#x2062;1ln2

Question 12

If y=lnxx2, what is dydx?

Accepted Answer

1-2lnxx3

Question 13

Which of the following is the correct simplification for ln(e-xcos2xx2+x+1)?

Accepted Answer

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

Question 14

Which of the following is the correct derivative for y=x+aln(cosx)?

Accepted Answer

1-atanx

Question 15

If y=(sinx)lnx, which of the following is the correct expression for lny?

Accepted Answer

lny=lnx&#x2062;ln(sinx)

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

Study Guide - Practice Questions