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

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

Question 1

Let $$y = rac{	an x}{x}. $$Which of the following is the correct expression for $$y'$$?

Accepted Answer

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

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Given $$y = \ln \left( \frac{e$$^{-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 6

For $$y = \ln \left( \frac{e$$^{-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 7

A biologist models the population of a bacteria culture by $$P(t) = (2t+1$$)^{\ln t}$$ for t$$ \u003e 0$$. What is P$$'(t)$$?

Accepted Answer

$$P'(t) = (2t+1$$)^{\ln t}$$ \left( \frac{2}{2t+1} \ln t + \frac{1}{t} \ln(2t+1) \right)$$

Question 8

A chemical reaction's rate is given by $$R(x) = \ln(x^2$ + 1$$)^{e^{-x}$$}. $$What is $$R'(x)$$?

Accepted Answer

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

Question 9

Suppose $$y = \ln(x^2$ + x + 1). $$Which of the following is $$y'$$?

Accepted Answer

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

Question 10

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = \ln(\sin t$$)^{\ln t}$$ for t$$ \u003e 0$$. What is C$$'(t)$$?

Accepted Answer

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

Question 11

Let $$y = \ln(x^2$ + x + 1). If$$ $$y' = \frac{2x + 1}{x^2$ + x + 1}$$, what is $$y''$$?

Accepted Answer

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

Question 12

A physicist models the displacement of a particle as $$s(t) = \ln(e$$^{-t}$$ \cos^2$ t). $$What is $$s'(t)$$?

Accepted Answer

$$s'(t) = -1 - 2 	an t$$

Question 13

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

Accepted Answer

$$y' = -2 	an x$$

Question 14

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

Accepted Answer

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

Question 15

Suppose $$y = \ln(x^2$ + x + 1). $$Which of the following best describes the concavity of $$y$$ for $$x \u003e 0$$?

Accepted Answer

$$y is $$concave down for all $$x \u003e 0$$

Question 16

A function is defined as $$f(x) = \log_2$(e^{-x}$$ \cos(\pi x))$$. For which x$ is f(x$$)$ undefined?

Accepted Answer

$$x$$ such that $$\cos(\pi x) = 0$$

Question 17

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

Accepted Answer

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

Question 18

Suppose $$y = \ln(x^2$ + x + 1). $$What is the value of $$y' at$$ $$x = 0$$?

Accepted Answer

$$y'(0) = 1$$

Advanced Differentiation Techniques in Calculus: Logarithmic and Exponential Functions

Study Guide - Practice Questions