Q: Given $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right)$$, what is $$y'$$?

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

Question 1

Let $$y = \log_2$ \left( e$$^{-x}$$ \cos(\pi x) \right). $$What is $$y'$$?

Accepted Answer

$$y' = \frac{-e$$^{-x}$$ \cos(\pi x) - e$$^{-x}$$ \sin(\pi x) \pi}{e$$^{-x}$$ \cos(\pi x) \ln 2}$$

Question 2

Given $$y = \frac{\ln x}{x^2$}$$, what is $$y''$$?

Accepted Answer

$$y'' = \frac{x^2$ (1 - 2 \ln x + 2) - (x - 2 \ln x) \cdot 4x}{x^4$}$$

Question 3

Let $$y = (\sin x$$)^{\ln x}$$. What is y$$'$$?

Accepted Answer

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

Question 4

Suppose $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right). $$Which of the following is the correct simplification for $$y$$?

Accepted Answer

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

Question 5

Given $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right)$$, what is $$y'$$?

Accepted Answer

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

Question 6

Which of the following is the correct application of the product rule for $$f(x) = x \ln(\sin x)$$?

Accepted Answer

$$f'(x) = \ln(\sin x) + x \cot x$$

Question 7

If $$y = \log_2$ (e$$^{-x}$$ \cos(\pi x))$$, which of the following best describes the steps to find $$y'$$?

Accepted Answer

Use the chain rule and the properties of logarithms, then differentiate the inside function and divide by the original argument times $$\ln 2.$$

Question 8

A biologist models the population of a bacteria culture by $$P(t) = (\sin t$$)^{\ln t}$$ for t$$ \u003e 0$$. What is the instantaneous rate of change of the population at time t$?

Accepted Answer

$$P'(t) = (\sin t$$)^{\ln t}$$ \left( \frac{\cot t}{t} \right)$$

Question 9

Given $$y = \frac{\ln x}{x^2$}$$, which of the following is the correct first derivative $$y'$$?

Accepted Answer

$$y' = \frac{1 - 2 \ln x}{x^3$}$$

Question 10

A chemical reaction's rate is modeled by $$R(x) = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right). $$What is the effect on $$R'(x) as$$ $$x$$ increases, assuming $$x \u003e 0$$?

Accepted Answer

$$R'(x)$$ becomes more positive due to the $$\frac{2x+1}{x^2$ + x + 1}$$ term dominating as $$x$$ increases.

Question 11

Which of the following is the correct logarithmic differentiation step for $$y = (\sin x$$)^{\ln x}$$?

Accepted Answer

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

Question 12

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

Accepted Answer

$$y' = \frac{-e$$^{-x}$$ \cos(\pi x) - e$$^{-x}$$ \sin(\pi x) \pi}{e$$^{-x}$$ \cos(\pi x) \ln 2}$$

Question 13

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = \frac{\ln t}{t^2$}$$ for $$t \u003e 0. $$What is the rate of change of concentration at time $$t$$?

Accepted Answer

$$C'(t) = \frac{1 - 2 \ln t}{t^3$}$$

Question 14

Which of the following best describes the use of logarithmic properties to simplify $$\ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right)$$?

Accepted Answer

Expand as $$-x + 2 \ln(\cos x) - \ln(x^2$ + x + 1)$$

Question 15

Suppose $$y = (\sin x$$)^{\ln x}$$. Which of the following is the correct step to find y$$'$$ using logarithmic differentiation?

Accepted Answer

Take $$\ln of $$both sides, then differentiate implicitly to get $$\frac{y'}{y} = \frac{\cot x}{x}$$

Question 16

If $$y = \frac{\ln x}{x^2$}$$, which of the following is the correct application of the quotient rule for $$y'$$?

Accepted Answer

$$y' = \frac{x^2$ \cdot \frac{1}{x} - \ln x \cdot 2x}{x^4$}$$

Question 17

A physicist models the amplitude of a wave as $$A(x) = (\sin x$$)^{\ln x}$$. What is the amplitude's rate of change with respect to x$?

Accepted Answer

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

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

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