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

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 $a > 0$?

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

Study Guide - Practice Questions