Question 1

Which of the following correctly states the Product Rule for differentiation?

Accepted Answer

$(f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x)$

Question 2

Given $y = (8x^2 - 1)(2x - 5)$, what is $y'$?

Accepted Answer

$y' = 48x^2 - 80x - 2$

Question 3

A pharmaceutical company models the concentration of a drug in the bloodstream as $C(t) = t^2 e^t - 17t$. What is the rate of change of concentration at time $t$?

Accepted Answer

$C'(t) = 2t e^t + t^2 e^t - 17$

Question 4

Which of the following is the correct application of the Quotient Rule for $y = \frac{4x - 5}{x + 3}$?

Accepted Answer

$y' = \frac{17}{(x + 3)^2}$

Question 5

A population model is given by $P(x) = \frac{7e^x}{3e^x + 8}$. What is $P'(x)$?

Accepted Answer

$P'(x) = \frac{56e^x}{(3e^x + 8)^2}$

Question 6

If $y = (4t - 1)(7t + 7)^{-1}$, what is $y'$?

Accepted Answer

$y' = \frac{35}{(7t + 7)^2}$

Question 7

A function is given by $y = \frac{x - 6}{x e^x}$. What is the derivative $y'$?

Accepted Answer

$y' = \frac{-x^2 e^x + 6e^x + 6x e^x}{x^2 e^{2x}}$

Question 8

Which of the following is the correct formula for the Quotient Rule?

Accepted Answer

$\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

Question 9

A biologist models the growth rate of bacteria as $B(x) = x^2 e^x$. What is $B'(x)$?

Accepted Answer

$B'(x) = 2x e^x + x^2 e^x$

Question 10

Find the equation of the tangent line to $f(x) = \frac{3x}{x+1}$ at $x = 2$.

Accepted Answer

$y = \frac{1}{3}x + \frac{4}{3}$

Question 11

Which rule should be used to differentiate $y = x^2 e^x - 17x$?

Accepted Answer

Product Rule and Sum/Difference Rule

Question 12

A chemical reaction rate is modeled by $R(x) = \frac{3x}{x+1}$. What is $R'(x)$?

Accepted Answer

$R'(x) = \frac{3}{(x+1)^2}$

Question 13

If $f(x) = x^2$ and $g(x) = e^x$, what is the derivative of $f(x)g(x)$?

Accepted Answer

$2x e^x + x^2 e^x$

Question 14

Which of the following is NOT a common mistake when applying the Product Rule?

Accepted Answer

Using the Quotient Rule for a product

Question 15

A function is given by $y = \frac{x - 6}{x e^x}$. Which rule(s) must be used to find $y'$?

Accepted Answer

Quotient Rule and Product Rule

Question 16

If $y = (4t - 1)(7t + 7)^{-1}$, which real-world scenario could this model?

Accepted Answer

The concentration of a substance inversely proportional to another changing quantity

Question 17

Which of the following best describes the difference between the Product Rule and the Quotient Rule?

Accepted Answer

Product Rule is used for multiplication, Quotient Rule for division of functions

Question 18

A function $f(x)$ is defined as $f(x) = x^2 e^x$. Which of the following is the correct interpretation of $f'(x)$ in a real-world context?

Accepted Answer

The instantaneous rate of change of the modeled quantity at time $x$

3.4: The Product and Quotient Rules – Calculus Study Notes

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