Q: Let $$y = rac{ an x}{x}. $$What is $$y'$$?

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

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

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

Q: Suppose $$y = x \ln(\cos x). $$What is $$y'$$?

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

Q: Given $$y = x \ln(\cos x)$$, which of the following is the correct application of the product rule?

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

Question 1

Let $$y = rac{	an x}{x}. $$What is $$y'$$?

Accepted Answer

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

Question 2

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 3

If $$y = \ln x^x$$, what is $$y'$$?

Accepted Answer

$$y' = 1 + \ln x$$

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

Let $$y = \ln(e$$^{-x}$$ \cos^2$ x). $$Which of the following is equivalent to $$y$$?

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = \ln(t^2$ + t + 1)$$, where $$t is $$time in hours. What is the rate of change of concentration at $$t = 2$$ hours?

Accepted Answer

$$C'(2) = \frac{5}{7}$$

Question 12

A population grows according to $$P(t) = (\sin t$$)^{\ln t}$$ for t$$ \u003e 0$$. What is P$$'(t)$$?

Accepted Answer

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

Question 13

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 derivative of the logarithm base 2, then apply the quotient and product rules to the argument.

Question 14

Given $$y = x \ln(\cos x)$$, which of the following is the correct application of the product rule?

Accepted Answer

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

Question 15

A function is defined as $$f(x) = \ln(x^2$ + x + 1). $$Which of the following statements is true about $$f'(x)$$?

Accepted Answer

$$f'(x) is $$always positive for all real $$x.$$

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

$$y' = -2 	an x$$

Question 18

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

Accepted Answer

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

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

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