Q: Given \ln y = -x + 2 \ln(\cos x) - \$$ln(x^2 + x + 1$$), what is the correct expression for y' after differentiating both sides and solving for y'?

y' = y \left(-1 + 2 \tan x - \frac{2x + 1}{$$x^2 + x + 1$$}\right)

Q: If y = \frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$}, what is the derivative y' in terms of y and x?

y' = y \left(-1 + 2 \tan x - \frac{2x + 1}{$$x^2 + x + 1$$}\right)

Question 1

Which of the following functions is BEST suited for logarithmic differentiation?

Accepted Answer

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

Question 2

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

Accepted Answer

Rewrite using the change of base formula: log_a(u) = ln u / ln a

Question 3

Suppose y = [f(x)]^{g(x)}. Which of the following correctly expresses ln y in terms of f(x) and g(x)?

Accepted Answer

ln y = g(x) \cdot ln(f(x))

Question 4

For the function y = (sin $$x)^{ln x}$$, what is the derivative y'?

Accepted Answer

y' = (sin $$x)^{ln x}$$ \left( \frac{1}{x} ln(sin x) + ln x \cot x \right)

Question 5

A biologist models the growth of a bacteria colony with y = $$(2x^2$$ + $$1)^{\sqrt{x}$$}. Which differentiation technique is most efficient for finding dy/dx?

Accepted Answer

Logarithmic differentiation

Question 6

Given y = \frac{\ln x}{$$x^2$$}, what is the correct expression for the first derivative y'?

Accepted Answer

y' = \frac{x - 2x \ln x}{$$x^3$$}

Question 7

A physicist is analyzing a function y = \$$log_2$(3x^2$$ $$e^{-x}). $$What is the derivative y' with respect to x?

Accepted Answer

y' = \frac{1}{\ln 2} \left( \frac{6x - $$3x^2$$}{$$3x^2$$} - 1 \right)

Question 8

Which of the following is a correct application of the quotient rule for y = \frac{u}{v}?

Accepted Answer

y' = \frac{u'v - uv'}{$$v^2$$}

Question 9

Given \ln y = -x + 2 \ln(\cos x) - \$$ln(x^2 + x + 1$$), what is the correct expression for y' after differentiating both sides and solving for y'?

Accepted Answer

y' = y \left(-1 + 2 	an x - \frac{2x + 1}{$$x^2 + x + 1$$}ight)

Question 10

A medical researcher models the concentration of a drug in the bloodstream as y = \frac{$$e^{-kt}$$ \$$cos^2 t$$}{$$t^2 + 1$$}. Using logarithmic properties, which is the correct first step to simplify \ln y?

Accepted Answer

ln y = -kt + 2 \ln(\cos t) - \$$ln(t^2 + 1)$$

Question 11

If y = \log_a(u), what is the correct formula for y'?

Accepted Answer

y' = \frac{u'}{u \ln a}

Question 12

A chemist is studying a reaction rate modeled by y = $$(x^2$$ + $$1)^{\sin x}. $$What is the derivative y' with respect to x?

Accepted Answer

y' = $$(x^2$$ + $$1)^{\sin x}$$ \left( \frac{2x}{$$x^2 + 1$$} \sin x + \cos x \$$ln(x^2 + 1$$) \right)

Question 13

Which of the following is NOT a correct use of logarithmic properties when simplifying \ln\left(\frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$}\right)?

Accepted Answer

ln y = -x + 2 \cos x - \$$ln(x^2 + x + 1)$$

Question 14

A student is asked to find the second derivative of y = \frac{\ln x}{$$x^2$$}. Which of the following is the correct first step?

Accepted Answer

Apply the quotient rule to y'

Question 15

Which of the following real-world scenarios would most likely require the use of logarithmic differentiation?

Accepted Answer

Finding the rate of change of a function y = (1 + $$r/n)^{nt} in $$compound interest calculations

Question 16

If y = \frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$}, what is the derivative y' in terms of y and x?

Accepted Answer

y' = y \left(-1 + 2 	an x - \frac{2x + 1}{$$x^2 + x + 1$$}ight)

Question 17

Which of the following best describes the main advantage of logarithmic differentiation?

Accepted Answer

It simplifies the differentiation of products, quotients, and variable exponents.

Advanced Differentiation Techniques in Calculus

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