$$x^{\tan x}$$ \left( \$$sec^2 x$$ \ln x + \frac{\tan x}{x} \right)$$

Question 1

Which of the following is the correct derivative of $$y = \ln(4x)$$ with respect to $$x$$?

Accepted Answer

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

Question 2

A biologist models the population of bacteria as $$y = 3^x. $$What is the rate of change of the population with respect to time $$x$$?

Accepted Answer

$$y' = 3^x \ln 3$$

Question 3

Given $$y = \ln|\sec x|$$, which of the following is the correct derivative $$y'$$?

Accepted Answer

$$y' = 	an x$$

Question 4

A chemical reaction rate is modeled by $$y = \ln(10x). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x}$$

Question 5

If $$y = x \ln(2x)$$ models the entropy of a system, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\ln(2x) + 1$$

Question 6

A pharmaceutical company models the concentration of a drug as $$y = \ln((x+9)^2$ (3x+2)^3$). $$What is the derivative $$y'$$?

Accepted Answer

$$\frac{2}{x+9} + \frac{3}{3x+2}$$

Question 7

Given $$y = \ln\left(\frac{3x+1}{2x-5}\right)$$, what is $$y'$$?

Accepted Answer

$$\frac{3}{3x+1} - \frac{2}{2x-5}$$

Question 8

Which of the following is the correct derivative of $$y = \log_3$ x$$?

Accepted Answer

$$\frac{1}{x \ln 3}$$

Question 9

A population grows according to $$y = x$$^{	an x}$$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$x^{	an x}$$ \left( \$$sec^2 x$$ \ln x + \frac{	an x}{x} ight)$$

Question 10

Which rule allows you to find the derivative of $$y = \ln(u(x))$$?

Accepted Answer

Chain rule for logarithmic derivative: $$\frac{1}{u} u'$$

Question 11

A scientist uses $$y = \ln(a^b) in a $$calculation. Which logarithmic property simplifies this expression?

Accepted Answer

$$\ln(a^b) = b \ln a$$

Question 12

If $$y = \ln(ab)$$, which property of logarithms allows you to write $$y = \ln a + \ln b$$?

Accepted Answer

Product rule for logarithms

Question 13

A medical researcher models the spread of a virus as $$y = \ln(2x). $$What is the instantaneous rate of change at $$x = 1$$?

Accepted Answer

$$y' = 1$$

Question 14

Which of the following is the correct derivative of $$y = \ln|x|$$?

Accepted Answer

$$\frac{1}{x}$$

Question 15

A physicist models energy as $$y = \ln\left(\frac{a}{b}\right). $$Which property of logarithms is used to simplify this expression?

Accepted Answer

Quotient rule: $$\ln\left(\frac{a}{b}\right) = \ln a - \ln b$$

Question 16

If $$y = \ln(10x)$$, what is the value of $$y' at$$ $$x = 2$$?

Accepted Answer

$$\frac{1}{2}$$

Question 17

Which of the following is the correct derivative of $$y = a^x$$ for any constant $$a \u003e 0$$?

Accepted Answer

$$a^x \ln a$$

Question 18

A financial analyst models compound interest as $$y = \log_b x. $$What is the derivative $$y'$$?

Accepted Answer

$$\frac{1}{x \ln b}$$

Question 19

Which of the following is the correct derivative of $$y = \ln(u(x))$$ using the chain rule?

Accepted Answer

$$\frac{1}{u(x)} u'(x)$$

Derivatives of Logarithmic and Exponential Functions: 3.9

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