Q: If y = \ln \left( \frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$} \right ), what is the correct expression for y' in terms of y?

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

Question 1

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

Accepted Answer

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

Question 2

Given y = $$log_2$(e^{-x}$$ \, cos(\pi 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: y = \frac{\$$ln(e^{-x}$$ \, cos(\pi x))}{\ln 2}

Question 3

Suppose y = \ln \frac{x}{$$x^2$$}. What is the simplified form of y before differentiating?

Accepted Answer

y = -\ln x

Question 4

For the function y = (\sin $$x)^{\ln x}$$, after taking the natural logarithm of both sides, which expression do you obtain?

Accepted Answer

\ln y = \ln x \cdot \ln(\sin x)

Question 5

Given y = \ln \left( \frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$} \right ), which of the following correctly expands the logarithm using properties of logarithms?

Accepted Answer

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

Question 6

A biologist models the growth of a bacteria population with P(t) = $$(2t^2$$ + $$1)^{\sqrt{t}$$}. Which differentiation technique is most appropriate to find dP/dt?

Accepted Answer

Logarithmic differentiation

Question 7

If y = \ln \frac{x}{$$x^2$$}, what is the value of y'' (the second derivative) with respect to x?

Accepted Answer

y'' = \frac{1}{$$x^2$$}

Question 8

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

Accepted Answer

Because it allows us to turn exponents into products, making differentiation manageable.

Question 9

Given y = \$$log_2$(e^{-x}$$ \cos(\pi x)), which of the following is the correct expression for y'?

Accepted Answer

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

Question 10

A medical researcher is analyzing a drug concentration modeled by C(t) = $$(e^{-kt}$$ \$$cos^2$$ $$t)/(t^2 + 1). To $$find the rate of change of concentration, which differentiation strategy should be used?

Accepted Answer

Use logarithmic differentiation to expand and then differentiate

Question 11

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

Accepted Answer

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

Question 12

Which property of logarithms allows you to write \ln(a^b) as b \ln a?

Accepted Answer

Power property

Question 13

A chemist is studying a reaction rate given by R(x) = \ln \left( \frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$} \right ). After differentiating, which of the following terms appears in the derivative?

Accepted Answer

2 	an x

Question 14

If y = \ln \left( \frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$} \right ), what is the correct expression for y' in terms of y?

Accepted Answer

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

Question 15

Which of the following is NOT a correct property of logarithms?

Accepted Answer

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

Question 16

A student is asked to differentiate y = \$$ln(x^2$$ \sin x). Which of the following is the best first step?

Accepted Answer

Rewrite as y = \ln $$x^2$$ + \ln(\sin x)

Question 17

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

Accepted Answer

Finding the rate of change of a population modeled by P(t) = (3t + $$1)^{\sqrt{t}$$}

Advanced Differentiation Techniques in Calculus: Logarithmic Differentiation and Derivatives of Complex Functions

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