Q: A pharmaceutical company models the concentration of a drug in the bloodstream as y = ( ext{cos } $$x)^{ ext{ln } x$$}. What is the derivative y' with respect to x?

( ext{cos } $$x)^{ ext{ln } x$$} igg( rac{1}{x} ext{ln } ( ext{cos } x) - ext{ln } x an x igg)

Q: Given y = [g(x)]^{h(x)}, which of the following is the correct expression for y' using logarithmic differentiation?

y' = [g(x)]^{h(x)} igg( rac{h(x) g'(x)}{g(x)} + h'(x) ext{ln } g(x) igg)

Q: A medical researcher is studying a function y = ( ext{sin } $$x)^{ ext{ln } x$$}. What is the correct derivative y'?

( ext{sin } $$x)^{ ext{ln } x$$} igg( rac{1}{x} ext{ln } ( ext{sin } x) + ext{ln } x ext{cot } x igg)

Q: A function is given by y = ext{log}_$$2(e^{-x}$$ ext{cos}( ext{π}x)). Which of the following is the correct derivative y'?

y' = rac{1}{$$e^{-x}$$ ext{cos}( ext{π}x) ext{ln } 2} igg( $$-e^{-x}$$ ext{cos}( ext{π}x) + $$e^{-x} ( - $$ext{sin}( ext{π}x) ext{π} ) igg)

Question 1

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

Accepted Answer

y = [$$x^2 + 1$$]^{	an x}

Question 2

Suppose a biologist models the growth of a bacteria population with the function y = $$(e^{kt}$)^{rac{1}$${t}}, where k is a constant and t is time. Using logarithmic differentiation, what is the correct first step?

Accepted Answer

Take the natural logarithm of both sides: ln y = rac{1}{t} 	imes kt.

Question 3

Given y = rac{	an x}{$$x^2$$}, which differentiation rules must be used to find y'?

Accepted Answer

Quotient rule and chain rule

Question 4

A pharmaceutical company models the concentration of a drug in the bloodstream as y = (	ext{cos } $$x)^{	ext{ln } x$$}. What is the derivative y' with respect to x?

Accepted Answer

(	ext{cos } $$x)^{	ext{ln } x$$} igg( rac{1}{x} 	ext{ln } (	ext{cos } x) - 	ext{ln } x 	an x igg)

Question 5

Which of the following is the correct derivative of y = rac{	ext{ln } x}{$$x^2$$}?

Accepted Answer

y' = rac{1 - 2 	ext{ln } x}{$$x^3$$}

Question 6

A physicist is analyzing the function y = 	ext{log}_$$2(e^{-x}$$ 	ext{cos}(	ext{π}x)). What is the main reason to use the change of base formula before differentiating?

Accepted Answer

To convert the logarithm to a base that is easier to differentiate, such as the natural logarithm.

Question 7

Given y = [g(x)]^{h(x)}, which of the following is the correct expression for y' using logarithmic differentiation?

Accepted Answer

y' = [g(x)]^{h(x)} igg( rac{h(x) g'(x)}{g(x)} + h'(x) 	ext{ln } g(x) igg)

Question 8

A chemical reaction rate is modeled by y = rac{$$e^{-x}$$ 	ext{cos}^2 x}{$$x^2 + x + 1$$}. After taking the natural logarithm and differentiating, which of the following appears in the expression for rac{y'}{y}?

Accepted Answer

-1 + 2 	an x - rac{2x + 1}{$$x^2 + x + 1$$}

Question 9

Which property of logarithms allows you to rewrite ln(a^k) as k ln a before differentiating?

Accepted Answer

Power property

Question 10

A medical researcher is studying a function y = (	ext{sin } $$x)^{	ext{ln } x$$}. What is the correct derivative y'?

Accepted Answer

(	ext{sin } $$x)^{	ext{ln } x$$} igg( rac{1}{x} 	ext{ln } (	ext{sin } x) + 	ext{ln } x 	ext{cot } x igg)

Question 11

Which of the following is the correct application of the product rule for differentiating y = x 	ext{ln } x?

Accepted Answer

y' = 1 	imes 	ext{ln } x + x 	imes rac{1}{x}

Question 12

A function is given by y = rac{	ext{ln } x}{$$x^2$$}. What is the correct expression for the second derivative y''?

Accepted Answer

y'' = rac{$$(x^3 (1 - 2 	$$ext{ln } x))' - (x - 2x 	ext{ln } x) $$3x^2$$}{$$x^6$$}

Question 13

Which of the following is the correct derivative of y = a^x, where a \u003e 0 and a ≠ 1?

Accepted Answer

y' = a^x 	ext{ln } a

Question 14

A function is given by y = 	ext{log}_$$2(e^{-x}$$ 	ext{cos}(	ext{π}x)). Which of the following is the correct derivative y'?

Accepted Answer

y' = rac{1}{$$e^{-x}$$ 	ext{cos}(	ext{π}x) 	ext{ln } 2} igg( $$-e^{-x}$$ 	ext{cos}(	ext{π}x) + $$e^{-x} ( -	$$ext{sin}(	ext{π}x) 	ext{π} ) igg)

Question 15

Which of the following is NOT a correct property of logarithms used to simplify before differentiating?

Accepted Answer

ext{ln}(a + b) = 	ext{ln } a + 	ext{ln } b

Question 16

A function is given by y = rac{$$e^{-x}$$ 	ext{cos}^2 x}{$$x^2 + x + 1$$}. After logarithmic differentiation, what is the final form of y'?

Accepted Answer

y' = igg(-1 + 2 	an x - rac{2x + 1}{$$x^2 + x + 1$$}igg) rac{$$e^{-x}$$ 	ext{cos}^2 x}{$$x^2 + x + 1$$}

Advanced Differentiation Techniques in Calculus

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