Question 1

Which of the following correctly states the Product Rule for the derivative of two differentiable functions $$f(x)$$ and $$g(x)$$?

Accepted Answer

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

Question 2

Suppose $$h(x) = x^2$ e^x. $$Using the product rule, what is $$h'(x)$$?

Accepted Answer

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

Question 3

If $$y = (3x^2$ + 2x)(x - 5)$$, what is $$\frac{dy}{dx}$$ using the product rule?

Accepted Answer

$$ \frac{dy}{dx} = (6x + 2)(x - 5) + (3x^2 + 2x)(1) $$

Question 4

Which of the following is the correct Quotient Rule for differentiable functions $$f(x)$$ and $$g(x)$$, where $$g(x) 
eq 0$$?

Accepted Answer

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

Question 5

Given $$f(x) = \frac{x^2$ + 1}{x}$$, find $$f'(x)$$ using the quotient rule.

Accepted Answer

$$ f'(x) = \frac{(2x)x - (x^2 + 1)(1)}{x^2} $$

Question 6

A company’s revenue $$R(t) in $$thousands of dollars is given by $$R(t) = t^2$ e$$^{0.1t}$$, where t$ is time in months. What is the rate of change of revenue at t = 5$ months?

Accepted Answer

$$ R'(5) = 2(5)e^{0.5} + (5^2)(0.1)e^{0.5} $$

Question 7

If $$y = \frac{2x + 3}{x^2$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$ \frac{dy}{dx} = \frac{2x^2 - (2x + 3)2x}{x^4} $$

Question 8

The total sales $$S (in $$thousands of games) of a video game $$t$$ months after introduction is $$S(t) = 10000 \left(1 - \frac{1}{t+1}\right). $$What is $$S'(t)$$?

Accepted Answer

$$ S'(t) = 10000 \left( \frac{1}{(t+1)^2} \right) $$

Question 9

For the function $$f(x) = (x^2$ + 1)(x - 3)$$, at what value(s) of $$x is $$the tangent line horizontal?

Accepted Answer

$$ x = 1 	ext{ and } x = 3 $$

Question 10

A company’s cost function is $$C(x) = x^2$ + 5x. If $$the price per unit is $$p(x) = 20 - x$$, what is the derivative of the revenue $$R(x) = x p(x)$$?

Accepted Answer

$$ R'(x) = (1)(20 - x) + x(-1) $$

Question 11

If $$y = (x^3$ - 2x)(x^2$ + 4)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$ \frac{dy}{dx} = (3x^2 - 2)(x^2 + 4) + (x^3 - 2x)(2x) $$

Question 12

A company’s profit function is $$P(x) = (50 - x)x - (20x + 100). $$What is $$P'(x)$$?

Accepted Answer

$$ P'(x) = (50 - x) + x(-1) - 20 $$

Question 13

If $$f(x) = x^2$ and g(x) = x + 1$, which of the following is $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$$?

Accepted Answer

$$ \frac{d}{dx}\left(\frac{x^2}{x+1}\right) = \frac{2x(x+1) - x^2(1)}{(x+1)^2} $$

Question 14

A company’s marginal cost is the derivative of its cost function. If $$C(x) = 5x^2$ + 3x + 10$$, what is the marginal cost when $$x = 4$$?

Accepted Answer

$$ C'(4) = 10(4) + 3 = 43 $$

Question 15

If $$y = \frac{3x^2$ + 2x}{x^3$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$ \frac{dy}{dx} = \frac{(6x + 2)x^3 - (3x^2 + 2x)3x^2}{x^6} $$

Question 16

A company’s total sales $$S(t) (in $$thousands) after $$t$$ months is $$S(t) = 10000 \left(1 - \frac{1}{t+1}\right). If $$after 10 months, $$S(10) = 9000$$ and $$S'(10) = 1000$$, estimate the total sales after 11 months.

Accepted Answer

$$ S(11) \approx S(10) + S'(10) = 9000 + 1000 = 10000 $$

Derivatives of Products and Quotients – Business Calculus Study Notes

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