Question 1

Which of the following correctly represents 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

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

Accepted Answer

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

Question 4

Find the derivative of $$ y = x^2$ e^x - 17x .$$

Accepted Answer

$$ y' = 2x e^x + x^2$ e^x - 17 $$

Question 5

Given $$ y = \frac{4x - 5}{x + 3} $$, what is $$ y' $$?

Accepted Answer

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

Question 6

A pharmaceutical company models the concentration of a drug in the bloodstream as $$ C(t) = (4t - 1)/(7t + 7) . $$What is the rate of change of concentration at any time $$ t $$?

Accepted Answer

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

Question 7

Find the derivative of $$ y = \frac{7e^x}{3e^x + 8} .$$

Accepted Answer

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

Question 8

A population model is given by $$ P(x) = \frac{3x}{x + 1} . $$What is the slope of the tangent line to $$ P(x)  at$$ $$ x = 2 $$?

Accepted Answer

$$ 1/3 $$

Question 9

Which of the following is the correct derivative for $$ y = \frac{x - 6}{x e^x} $$?

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

A chemical reaction rate is modeled by $$ R(x) = (x^2$ - 1)(2x + 3) . $$What is $$ R'(x) $$?

Accepted Answer

$$ R'(x) = 2x(2x + 3) + (x^2$ - 1) \cdot 2 $$

Question 12

Which rule should be used to differentiate $$ y = (x^2$ + 1)(x^3$ - 2x) $$?

Accepted Answer

Product Rule

Question 13

Which rule should be used to differentiate $$ y = \frac{5x^2$ - 3}{2x + 1} $$?

Accepted Answer

Quotient Rule

Question 14

A medical dosage function is $$ D(x) = (x^2$ + 4x)(e^x) . $$What is $$ D'(x) $$?

Accepted Answer

$$ D'(x) = (2x + 4)e^x + (x^2$ + 4x)e^x $$

Question 15

If $$ f(x) = 3x $$ and $$ g(x) = x + 1 $$, what is the equation of the tangent line to $$ f(x)/g(x)  at$$ $$ x = 2 $$?

Accepted Answer

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

Question 16

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

Accepted Answer

Multiplying the derivatives instead of adding them

Study Notes: The Product and Quotient Rules in Calculus

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