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

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

Q: If $$y = x \ln(\cos x)$$, what is $$y'$$?

$$y' = \ln(\cos x) + x \cdot (-\tan x)$$

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

$$y'' = \frac{x^2$(1 - 2\ln x) - (x - 2x \ln x) \cdot 4x}{x^6$}$$

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

If $$y = \ln x$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\frac{d^2$y}{dx^2$} = -\frac{1}{x^2$}$$

Question 6

A chemical reaction rate is modeled by $$R(x) = \ln(x^2$ + x + 1). $$What is the instantaneous rate of change of $$R at$$ $$x = 1$$?

Accepted Answer

$$R'(1) = \frac{2 \cdot 1 + 1}{1^2$ + 1 + 1}$$

Question 7

Let $$y = \ln(\cos x). $$What is $$y'$$?

Accepted Answer

$$y' = -	an x$$

Question 8

If $$y = x \ln(\cos x)$$, what is $$y'$$?

Accepted Answer

$$y' = \ln(\cos x) + x \cdot (-	an x)$$

Question 9

A population model is given by $$P(x) = e$$^{-x}$$ \cos(nx). $$What is $$P'(x)$$?

Accepted Answer

$$P'(x) = -e$$^{-x}$$ \cos(nx) - n e$$^{-x}$$ \sin(nx)$$

Question 10

Let $$y = x + a \ln(\cos x). $$What is $$y'$$?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

A medical dosage is modeled by $$D(x) = \ln(x^2$ + x + 1). If $$the dosage increases at a rate proportional to $$D'(x)$$, what is the proportionality constant if $$D'(x) = \frac{2x + 1}{x^2$ + x + 1}$$?

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

If $$y = \ln(\sin x$$)^{\ln x}$$, what is $$\frac{1}{y} y'$$?

Accepted Answer

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

Question 17

Suppose $$y = x \ln(\sin x). $$What is $$y'$$?

Accepted Answer

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

Question 18

A function $$f(x) = \ln(x^2$ + x + 1)$$ models the growth of bacteria. What is the rate of change of growth at $$x = 0$$?

Accepted Answer

$$f'(0) = 1$$

Advanced Differentiation Techniques in Calculus

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