Question 1

Which of the following functions would benefit most from logarithmic differentiation?

Accepted Answer

y = $$(x^2$$ + $$1)^{	an x}$$

Question 2

Given y = $$log_2$(e^{-x} cos$$(πx)), which of the following is the correct first step in differentiating y with respect to x?

Accepted Answer

Rewrite $$log_2 as ln $$divided by ln 2

Question 3

Suppose y = (sin $$x)^{ln x}. $$After taking the natural logarithm of both sides, which expression do you obtain?

Accepted Answer

ln y = ln x * ln(sin x)

Question 4

For the function y = $$ln((e^{-x}$$ $$cos^2$$ $$x)/(x^2 + x + 1$$)), which of the following is the correct simplification using logarithm properties before differentiating?

Accepted Answer

$$ln(e^{-x}$$) + $$ln(cos^2 x$$) - $$ln(x^2 + x + 1)$$

Question 5

A biologist models the growth of a bacteria population with P(t) = $$(3t^2$$ + $$1)^{	an t}. To $$find the rate of change of the population at time t, which differentiation technique is most appropriate?

Accepted Answer

Logarithmic differentiation

Question 6

Given y = ln x / $$x^2$$, what is the correct expression for y'?

Accepted Answer

y' = (x - 2x ln x) / $$x^3$$

Question 7

A physicist is analyzing a function f(x) = $$(e^{x^2} sin$$ $$x)^{x}. $$Which of the following is the correct first step for differentiating f(x)?

Accepted Answer

Take the natural logarithm of both sides

Question 8

Which of the following is the derivative of y = $$ln((e^{-x}$$ $$cos^2$$ $$x)/(x^2 + x + 1)) $$with respect to x?

Accepted Answer

y' = -1 - 2 tan x + (2x + $$1)/(x^2 + x + 1)$$

Question 9

If y = (sin $$x)^{ln x}$$, what is the correct expression for y' after applying logarithmic differentiation?

Accepted Answer

y' = (sin $$x)^{ln x}$$ [ln x cot x + ln(sin x)/x]

Question 10

A medical researcher is modeling the concentration of a drug in the bloodstream as C(t) = $$ln(t^2 + 1$$) / $$t^2. $$What is the correct expression for the rate of change of concentration with respect to time?

Accepted Answer

Question 11

Which of the following is NOT a correct property of logarithms used in logarithmic differentiation?

Accepted Answer

ln(a + b) = ln a + ln b

Question 12

A chemist is analyzing a reaction rate modeled by R(x) = $$(x^2$$ + $$1)^{e^x}. $$Which of the following correctly expresses ln R(x) for the purpose of logarithmic differentiation?

Accepted Answer

ln R(x) = e^x $$ln(x^2 + 1)$$

Question 13

Which differentiation rule is most appropriate for y = ln x / $$x^2$$?

Accepted Answer

Quotient rule

Question 14

Suppose y = $$log_2$(e^{-x} cos$$(πx)). Which of the following is the correct formula for dy/dx?

Accepted Answer

y' = [$$1/(e^{-x} cos$$(πx) ln 2)] * [$$-e^{-x} cos$$(πx) + $$e^{-x}$$ ( -π sin(πx) )]

Question 15

A student is asked to find the second derivative of y = ln x / $$x^2. $$Which of the following is the correct approach?

Accepted Answer

Apply the quotient rule to y' again

Question 16

Which of the following best explains why logarithmic differentiation is especially useful for functions of the form $$f(x)^{g(x)}$$?

Accepted Answer

It allows the exponent to be brought down as a coefficient after taking the logarithm.

Question 17

A function is given by y = $$ln((e^{-x}$$ $$cos^2$$ $$x)/(x^2 + x + 1)). $$Which of the following is the correct derivative of the term 2 ln(cos x) with respect to x?

Accepted Answer

-2 tan x

Question 18

Which of the following is the correct derivative of y = $$ln(x^2 + x + 1) $$with respect to x?

Accepted Answer

(2x + $$1)/(x^2 + x + 1)$$

Advanced Differentiation Techniques in Calculus: Logarithmic Differentiation and the Chain Rule

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