Question 1

Which of the following correctly applies the Product Rule for natural logarithms to $$\ln(5x)$$?

Accepted Answer

$$\ln(5x) = \ln 5 + \ln x$$

Question 2

What is the derivative of $$y = \ln(4x^3$ + 5x + 4e$$^{-2x}$$)$$ with respect to $$x$$?

Accepted Answer

$$\frac{12x^2$ + 5 - 8e$$^{-2x}$$}{4x^3$ + 5x + 4e$$^{-2x}$$}$$

Question 3

Which property of logarithms allows you to write $$\ln\left(\frac{b}{x}\right) = \ln b - \ln x$$?

Accepted Answer

Quotient Rule

Question 4

Find the derivative of $$y = \ln(3te$$^{-t}$$)$$ with respect to $$t.$$

Accepted Answer

$$\frac{1}{t} - 1$$

Question 5

If $$y = x^x$$, what is $$\frac{dy}{dx}$$ using logarithmic differentiation?

Accepted Answer

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

Question 6

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

Accepted Answer

$$-	an 	heta$$

Question 7

Given $$y = \sin x$$^{\ln x}$$, what is $$\frac{dy}{dx}$$ using logarithmic differentiation?

Accepted Answer

$$(\sin x$$)^{\ln x}$$ \left( \cot x \ln x + \frac{1}{x} \ln(\sin x) \right)$$

Question 8

Which of the following is the correct application of the Power Rule for logarithms?

Accepted Answer

$$\ln(x^r) = r \ln x$$

Question 9

A biologist models bacteria growth with $$N(t) = 2$$^{3t}$$. What is $$\frac{dN}{dt}$$?

Accepted Answer

$$2^{3t}$$ \cdot 3 \ln 2$$

Question 10

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

Accepted Answer

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

Question 11

If $$y = \ln\left(\frac{x+1}{2x+1}\$$right)^5$$, what is $$\frac{dy}{dx}$$ after expanding the logarithm?

Accepted Answer

$$5 \left( \frac{1}{x+1} - \frac{2}{2x+1} \right)$$

Question 12

Which property of logarithms is used to simplify $$\ln\left(\frac{1}{x}\right)$$?

Accepted Answer

Reciprocal Rule

Question 13

A medication decays in the bloodstream according to $$C(t) = 10e$$^{-0.5t}$$. What is $$\frac{dC}{dt}$$?

Accepted Answer

$$-5e^{-0.5t}$$

Question 14

Which of the following is the correct derivative of $$y = a^x$$ with respect to $$x$$?

Accepted Answer

$$a^x \ln a$$

Question 15

If $$y = \ln(3x^2$ + 2x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{6x + 2}{3x^2$ + 2x}$$

Question 16

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

Accepted Answer

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

Derivatives of Inverse Functions and Logarithms

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