Q: A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = (t^2$ + 1$$)^{rac{1}$${t}}. $$What is the derivative $$C'(t)$$ with respect to $$t$$?

$$C'(t) = (t^2$ + 1$$)^{1/t}$$ igg( rac{2t}{t^2$ + 1} imes rac{1}{t} - rac{rac{1}{t^2$} imes rac{1}{t^2$ + 1}}{1} igg)$$

Q: A chemical reaction rate is modeled by $$R(x) = ( ext{sin} x$$)^{ ext{ln}$$ x}. $$What is $$R'(x)$$?

$$R'(x) = ( ext{sin} x$$)^{ 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}$$ ext{cos}( ext{π}x)). $$What is $$y'$$?

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

Question 1

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

Accepted Answer

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

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = (t^2$ + 1$$)^{rac{1}$${t}}. $$What is the derivative $$C'(t)$$ with respect to $$t$$?

Accepted Answer

$$C'(t) = (t^2$ + 1$$)^{1/t}$$ igg( rac{2t}{t^2$ + 1} 	imes rac{1}{t} - rac{rac{1}{t^2$} 	imes rac{1}{t^2$ + 1}}{1} igg)$$

Question 3

Given $$y = rac{	an x}{e^x}$$, which differentiation rule(s) must be applied to find $$y'$$?

Accepted Answer

Quotient rule and chain rule

Question 4

A biologist models population growth as $$P(t) = e$$^{k 	an t}$$, where k$ is a constant. What is P$$'(t)$$?

Accepted Answer

Question 5

A correct answer is missing in the previous question. Please select the correct derivative for $$P(t) = e$$^{k 	an t}$$.

Accepted Answer

Question 6

Let $$y = rac{	an x}{e^x}. $$What is $$y'$$?

Accepted Answer

$$y' = rac{	an' x 	imes e^x - 	an x 	imes e^x}{e$$^{2x}$$}$$

Question 7

A medical researcher is analyzing the rate of change of a medication's effectiveness, modeled by $$E(x) = rac{	ext{ln}(x)}{x^2$}. $$What is $$E'(x)$$?

Accepted Answer

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

Question 8

Which of the following is the correct derivative of $$y = 	ext{ln}(e$$^{-x}$$ 	ext{cos}^2 x)$$?

Accepted Answer

$$y' = -1 - 2 	an x$$

Question 9

A chemical reaction rate is modeled by $$R(x) = (	ext{sin} x$$)^{	ext{ln}$$ x}. $$What is $$R'(x)$$?

Accepted Answer

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

Question 10

Which of the following statements about logarithmic differentiation is TRUE?

Accepted Answer

It is especially useful for functions with variable exponents or products/quotients of several functions.

Question 11

A function is given by $$y = 	ext{log}_2(e$$^{-x}$$ 	ext{cos}(	ext{π}x)). $$What is $$y'$$?

Accepted Answer

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

Question 12

Which of the following is the correct application of the chain rule for differentiating $$y = 	ext{ln}(g(x))$$?

Accepted Answer

$$y' = rac{1}{g(x)} 	imes g'(x)$$

Question 13

A historian models the spread of information as $$I(t) = a^t$$, where $$a \u003e 0. $$What is the derivative $$I'(t)$$?

Accepted Answer

$$I'(t) = a^t 	ext{ln} a$$

Question 14

Which of the following is the correct formula for the quotient rule?

Accepted Answer

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2$}$$

Question 15

A function is given by $$y = 	ext{ln} rac{e$$^{-x}$$ 	ext{cos}^2 x}{x^2$ + x + 1}. $$Which of the following is the correct first step to simplify $$y$$ before differentiating?

Accepted Answer

$$y = -x + 2 	ext{ln}(	ext{cos} x) - 	ext{ln}(x^2$ + x + 1)$$

Question 16

Which of the following is NOT a correct use of the product, quotient, or chain rule?

Accepted Answer

Using the quotient rule for $$y = \ln x \cdot x^2$

Question 17

A function is given by $$y = e$$^{f(x)}$$. Which of the following is the correct derivative?

Accepted Answer

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

Advanced Differentiation Techniques in Calculus: Logarithmic and Implicit Differentiation

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