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

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

Question 1

Given $$y = \log_2$(e^{-x}$$ \cos(\pi x))$$, what is y$$'$?

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

Given $$y = (\sin x$$)^{\ln x}$$, what is y$$'$$?

Accepted Answer

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

Question 4

For $$y = \ln \left( \frac{e$$^{-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

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

Accepted Answer

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

Question 6

Which of the following best describes the process used to differentiate $$y = (\sin x$$)^{\ln x}$$?

Accepted Answer

Logarithmic differentiation is used, taking $$\ln of $$both sides and then differentiating implicitly.

Question 7

Suppose a population $$P at $$time $$t is $$modeled by $$P(t) = \ln(t^2$ + 1). $$What is the instantaneous rate of change of the population at $$t=2$$?

Accepted Answer

$$P'(2) = \frac{4}{5}$$

Question 8

A chemical reaction's rate is given by $$R(x) = x e$$^{-x}$$. What is the value of R$$'(1)$$?

Accepted Answer

$$R'(1) = 0$$

Question 9

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

Accepted Answer

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

Question 10

A medication's concentration in the bloodstream is modeled by $$C(t) = \ln(5t + 1). $$What is the rate of change of concentration at $$t=2$$?

Accepted Answer

$$C'(2) = \frac{5}{11}$$

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

A function $$f(x) is $$defined as $$f(x) = \ln(\cos^2$ x). $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = -2 	an x$$

Question 14

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

Accepted Answer

Take $$\ln of $$both sides: $$\ln y = \ln x \ln(\sin x)$$

Question 15

If $$y = \ln(e$$^{-x}$$ \cos^2$ x)$$, what is $$y'$$?

Accepted Answer

$$y' = -1 - 2 	an x$$

Question 16

A real-world scenario: The brightness $$B of a $$star as seen from Earth is modeled by $$B(d) = \ln(\frac{1}{d^2$})$$, where $$d is $$the distance from Earth. What is $$B'(d)$$?

Accepted Answer

$$B'(d) = -\frac{2}{d}$$

Question 17

Given $$y = \ln(\cos^2$ x)$$, which of the following is a correct simplification before differentiating?

Accepted Answer

$$y = 2 \ln(\cos x)$$

Question 18

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

Accepted Answer

$$y'(1) = \frac{3}{3} = 1$$

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

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