Q: For $$y = \ln\left(\dfrac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1}\right)$$, which of the following is the correct expression for $$y'$$?

$$\dfrac{2x + 1}{x^2$ + x + 1} - 2 \tan x - 1$$

Question 1

Let $$y = \log_2$(e^{-x}$$ \cos(nx))$$. What is $$\dfrac{dy}{dx}$?

Accepted Answer

$$\dfrac{-e$$^{-x}$$ \cos(nx) + e$$^{-x}$$(-\sin(nx) n)}{e$$^{-x}$$ \cos(nx) \ln 2}$$

Question 2

Given $$y = \dfrac{\ln x}{x^2$}$$, find $$y''.$$

Accepted Answer

$$\dfrac{x^2$(1 - 2\ln x) - (x - 2\ln x) \cdot 4x}{x^4$}$$

Question 3

For $$y = \ln\left(\dfrac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1}\right)$$, which of the following is the correct expression for $$y'$$?

Accepted Answer

$$\dfrac{2x + 1}{x^2$ + x + 1} - 2 	an x - 1$$

Question 4

If $$y = (\sin x$$)^{\ln x}$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{\cot x}{x} (\sin x$$)^{\ln x}$$

Question 5

A biologist models the population of a species as $$P(t) = \ln(t^2$ + t + 1). $$What is the instantaneous rate of change of the population at $$t = 2$$?

Accepted Answer

$$\dfrac{2 \cdot 2 + 1}{2^2$ + 2 + 1}$$

Question 6

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

Accepted Answer

$$\dfrac{d}{dx} \log_2$(u) = \dfrac{1}{u \ln 2} \dfrac{du}{dx}$$, where $$u = e$$^{-x}$$ \cos(nx)$$

Question 7

A physicist models the intensity of a signal as $$I(x) = \dfrac{\ln x}{x^2$}. $$What is the value of $$I'(e)$$?

Accepted Answer

$$\dfrac{1 - 2 \ln e}{e^3$}$$

Question 8

Which of the following is the correct simplification for $$\ln(e$$^{-x}$$ \cos^2$ x)$$?

Accepted Answer

$$-x + 2 \ln(\cos x)$$

Question 9

If $$y = (\sin x$$)^{\ln x}$$, which logarithmic property is used to rewrite $$\ln y$$ for differentiation?

Accepted Answer

$$\ln(a^b) = b \ln a$$

Question 10

A chemical reaction rate is modeled by $$R(x) = \ln(x^2$ + x + 1). $$What is $$R'(x)$$?

Accepted Answer

$$\dfrac{2x + 1}{x^2$ + x + 1}$$

Question 11

Which of the following is the correct derivative of $$y = \ln x$$?

Accepted Answer

$$\dfrac{1}{x}$$

Question 12

If $$y = \ln(x^2$ + x + 1)$$, what is $$y''$$?

Accepted Answer

$$\dfrac{2(x^2$ + x + 1) - (2x + 1)^2$}{(x^2$ + x + 1)^2$}$$

Question 13

A medical researcher models the spread of a virus as $$V(x) = \log_2$(e^{-x}$$ \cos(nx))$$. What is the value of V$$'(0)$?

Accepted Answer

$$\dfrac{-1}{\ln 2}$$

Question 14

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

Accepted Answer

$$\dfrac{\cot x}{x} (\sin x$$)^{\ln x}$$

Question 15

If $$y = \dfrac{\ln x}{x^2$}$$, which rule is used to find $$y'$$?

Accepted Answer

Quotient rule

Question 16

A population model is given by $$P(x) = \ln(x^2$ + x + 1). $$What is the interpretation of $$P'(x)$$?

Accepted Answer

The instantaneous rate of change of the population with respect to $$x$$

Question 17

Which of the following is the correct simplification for $$\ln(e$$^{-x}$$ \cos^2$ x)$$?

Accepted Answer

$$-x + 2 \ln(\cos x)$$

Question 18

If $$y = \log_2$(e^{-x}$$ \cos(nx))$$, what is the domain of y$?

Accepted Answer

$$x$$ such that $$e^{-x}$$ \cos(nx) \u003e 0$$

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

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