Question 1

What is the derivative of $$y = 5 \ln x$$ with respect to $$x$$?

Accepted Answer

$$y' = \dfrac{5}{x}$$

Question 2

Given $$f(x) = x^3$ \ln x + 4x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = x^2$ + 3x^2$ \ln x + 4$$

Question 3

Which of the following is the correct formula for the derivative of $$\ln u$$ with respect to $$x$$?

Accepted Answer

$$\dfrac{d}{dx} \ln u = \dfrac{1}{u} \dfrac{du}{dx}$$

Question 4

Differentiate $$y = \ln(x^2$ - 5)$$ with respect to $$x.$$

Accepted Answer

$$y' = \dfrac{2x}{x^2$ - 5}$$

Question 5

If $$f(x) = \ln\left(\dfrac{x^3$ + 4}{x}\right)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{3x^2$}{x^3$ + 4} - \dfrac{1}{x}$$

Question 6

A psychological experiment models the percentage of students retaining syllables after $$t$$ minutes as $$R(t) = 80 - 27 \ln t. $$What is the rate of change of retention at $$t = 2$$ minutes?

Accepted Answer

$$R'(2) = -13.5$$

Question 7

For the function $$R(t) = 80 - 27 \ln t$$, at what time $$t$$ will the percentage of students retaining syllables drop to $$20\%$$?

Accepted Answer

$$t \approx 9.23$$

Question 8

Which of the following best describes the meaning of $$R'(t) in $$the context of the forgetting experiment?

Accepted Answer

The instantaneous rate at which the percentage of students retaining syllables is changing at time $$t$$

Question 9

Differentiate $$y = \ln(3x^2$ + 4)$$ with respect to $$x.$$

Accepted Answer

$$y' = \dfrac{6x}{3x^2$ + 4}$$

Question 10

Given $$y = \ln\left(\dfrac{x^5$ - 2}{x}\right)$$, what is $$y'$$?

Accepted Answer

$$y' = \dfrac{4x^5$ + 2}{x(x^5$ - 2)}$$

Question 11

Which rule is used to differentiate $$\ln(f(x))$$?

Accepted Answer

Chain rule

Question 12

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

Accepted Answer

$$y' = 2x \ln x + x$$

Question 13

Suppose $$y = \ln(5x)$$, what is $$y'$$?

Accepted Answer

$$y' = \dfrac{1}{x}$$

Question 14

Which of the following is the correct derivative of $$y = x^2$ - x^2$ \ln x$$?

Accepted Answer

$$y' = 2x - 2x \ln x - x$$

Question 15

A company models the depreciation of equipment value by $$V(t) = 100 - 15 \ln t$$ for $$t \geq 1. $$What is the instantaneous rate of depreciation at $$t = 5$$?

Accepted Answer

$$V'(5) = -3$$

Question 16

If $$f(x) = \ln(x^3$)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \dfrac{3}{x}$$

Question 17

Which of the following statements is true about the graph of $$R(t) = 80 - 27 \ln t$$?

Accepted Answer

It is a decreasing function for $$t \u003e 1$$

Question 18

If $$y = \ln(x^2$ + 4x)$$, what is $$y'$$?

Accepted Answer

$$y' = \dfrac{2x + 4}{x^2$ + 4x}$$

Question 19

A population model is given by $$P(t) = 200 - 40 \ln t. $$What is the population's rate of change at $$t = 4$$?

Accepted Answer

$$P'(4) = -10$$

Derivatives of Natural Logarithmic Functions and Applications in Business Calculus

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