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 $$y = \log_2$(e^{-x}$$ \cos(nx))$$. What is y$$'$$ with respect to x$?

Accepted Answer

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

Question 2

Given $$y = \frac{\ln x}{x^2$}$$, find $$y'' ($$the second derivative) with respect to $$x.$$

Accepted Answer

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

Question 3

For $$y = (\sin x$$)^{\ln x}$$, which of the following is the correct expression for y$$'$$?

Accepted Answer

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

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

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = \frac{\ln t}{t^2$}$$, where $$t is $$time in hours. What does $$C''(t)$$ represent in this context?

Accepted Answer

The rate at which the rate of change of concentration is changing

Question 7

If $$y = \log_2$(e^{-x}$$ \cos(nx))$$, which of the following is a necessary condition for y$ to be defined for real x$?

Accepted Answer

$$\cos(nx) 
eq 0$$

Question 8

Let $$y = (\sin x$$)^{\ln x}$$. For which values of x$ is y$ defined?

Accepted Answer

$$x \u003e 0$$ and $$\sin x \u003e 0$$

Question 9

A biologist models population growth with $$P(x) = \ln\left(\frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1}\right). $$What does the term $$-x in $$the simplified expression for $$P(x)$$ represent?

Accepted Answer

Exponential decay in population

Question 10

Given $$y = \frac{\ln x}{x^2$}$$, which of the following best describes the behavior of $$y as$$ $$x 	o \infty$$?

Accepted Answer

$$y 	o 0$$

Question 11

Suppose $$y = (\sin x$$)^{\ln x}$$. What is the value of y$ when x$$ = \frac{\pi}{2}$$?

Accepted Answer

$$y = 1$$

Question 12

For $$y = \log_2$(e^{-x}$$ \cos(nx))$$, which of the following is the correct domain for y$?

Accepted Answer

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

Question 13

A physicist models the intensity of a wave as $$I(x) = (\sin x$$)^{\ln x}$$. What does the derivative I$$'(x)$$ represent physically?

Accepted Answer

The rate of change of intensity with respect to position

Question 14

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

Accepted Answer

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

Question 15

If $$y = (\sin x$$)^{\ln x}$$, which of the following is the correct logarithmic form for y$?

Accepted Answer

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

Question 16

A chemical reaction rate is modeled by $$R(x) = \log_2$(e^{-x}$$ \cos(nx))$$. What does the derivative R$$'(x)$ tell you?

Accepted Answer

How quickly the reaction rate changes with respect to $$x$$

Question 17

Given $$y = \ln\left(\frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1}\right)$$, which term in the derivative $$y'$$ accounts for the oscillatory nature of the function?

Accepted Answer

$$-2 	an x$$

Question 18

For $$y = (\sin x$$)^{\ln x}$$, which of the following is the correct expression for $$\frac{y'}{y}$$?

Accepted Answer

$$\frac{\cot x}{x}$$

Advanced Differentiation Techniques in Calculus: Logarithmic and Exponential Functions

Study Guide - Practice Questions

Study Guide - Flashcards