Q: Given y = \ln\left(\frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$}\right), which of the following is the correct derivative y'?

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

Question 1

Which of the following functions is BEST suited for logarithmic differentiation?

Accepted Answer

y = $$(x^2$$ + $$1)^{	an x}$$

Question 2

Suppose y = (	an $$x)^{x^2}. $$Using logarithmic differentiation, what is the correct expression for $$ \frac{1}{y} \frac{dy}{dx} $$?

Accepted Answer

\frac{2x}{x} + $$x^2$$ \cot x

Question 3

Given y = \ln\left(\frac{$$e^{-x}$$ \$$cos^2 x$$}{$$x^2 + x + 1$$}\right), which of the following is the correct derivative y'?

Accepted Answer

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

Question 4

A biologist models the growth of a bacteria population with y = $$(3t^2$$ + $$1)^{\sin t}$$, where t is time in hours. What is the correct first step to find $$ \frac{dy}{dt} $$ using logarithmic differentiation?

Accepted Answer

Take the natural logarithm of both sides: \ln y = \sin t \cdot \$$ln(3t^2 + 1).$$

Question 5

If y = \frac{\ln x}{$$x^2$$}, what is the simplified form of y'?

Accepted Answer

y' = \frac{1 - 2 \ln x}{$$x^3$$}

Question 6

Which differentiation rule is most appropriate for finding the derivative of y = $$e^{x^2 \sin x}$$?

Accepted Answer

Chain rule

Question 7

A medical researcher models drug concentration as y = \$$log_2$(e^{-x}$$ \cos(\pi x)). What is the correct first step to differentiate y with respect to x?

Accepted Answer

Rewrite using the change of base formula: y = \frac{\$$ln(e^{-x}$$ \cos(\pi x))}{\ln 2}

Question 8

Given y = (\sin $$x)^{\ln x}$$, which of the following is the correct expression for y'?

Accepted Answer

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

Question 9

Which of the following statements about the chain rule is TRUE?

Accepted Answer

It is used to differentiate a composition of functions.

Question 10

A chemical reaction rate is modeled by y = \$$ln(e^{-kt}$$ \$$cos^2 t$$), where k \u003e 0. What is the derivative y' with respect to t?

Accepted Answer

y' = -k - 2 	an t

Question 11

Which of the following is the correct application of the product rule for y = $$x^2$$ \ln x?

Accepted Answer

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

Question 12

Suppose y = \frac{e^x}{\ln x}. Which rule(s) must be used to find y'?

Accepted Answer

Quotient rule and chain rule

Question 13

A physicist models a process with y = $$(x^2$$ + $$1)^{e^x}. $$Which of the following is the correct first step for logarithmic differentiation?

Accepted Answer

Take the natural logarithm: \ln y = e^x \$$ln(x^2 + 1)$$

Question 14

If y = \ln(f(x)), which of the following is the correct derivative?

Accepted Answer

y' = \frac{f'(x)}{f(x)}

Question 15

Which of the following best describes when to use the quotient rule?

Accepted Answer

When differentiating the division of two functions

Question 16

A function is given by y = $$e^{\ln(x^2 + 1)}. $$What is y'?

Accepted Answer

y' = 2x

Question 17

Which of the following is the correct derivative of y = $$e^{f(x)}$$?

Accepted Answer

y' = $$e^{f(x)} f$$'(x)

Question 18

A function is defined as y = \$$ln(x^2$$ \sin x). What is the correct derivative y'?

Accepted Answer

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

Question 19

Which of the following is NOT a correct step in logarithmic differentiation?

Accepted Answer

Apply the quotient rule to the original function before taking logarithms

Advanced Differentiation Techniques in Calculus

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