Question 1

Which of the following statements correctly describes the derivative of a constant function $$f(x) = c$$?

Accepted Answer

The derivative is $$f'(x) = 0$$

Question 2

Given $$f(x) = x^2$ and g(x) = x$, what is the derivative of h(x$$) = $$2x^2 - 3x + 5$?

Accepted Answer

$$h'(x) = 4x - 3$$

Question 3

If $$y = \frac{1}{x}$$, what is $$\frac{dy}{dx}$$ using the power rule?

Accepted Answer

$$\frac{dy}{dx} = -x$$^{-2}$$

Question 4

A business models its revenue as $$R(x) = 0.00027744x^3$ - 0.022137x^2$ + 0.726618x + 3.51722$$, where $$x is $$weeks since launch. What does $$R'(x)$$ represent in this context?

Accepted Answer

The rate of change of revenue with respect to weeks

Question 5

Find the derivative of $$y = -0.01x^2$ - 0.5x + 70.$$

Accepted Answer

$$y' = -0.02x - 0.5$$

Question 6

For $$f(x) = x^2$ - 6x + 10$$, what is the equation of the tangent line at the point $$(1, 5)$$?

Accepted Answer

$$y = -4x + 9$$

Question 7

If $$y = \sqrt{x}$$, what is $$\frac{dy}{dx}$$ using the power rule?

Accepted Answer

$$\frac{dy}{dx} = \frac{1}{2}x$$^{-1/2}$$

Question 8

For $$f(x) = x^3$ - 6x^2$ + 10$$, what is the slope of the tangent line at $$x = 1$$?

Accepted Answer

The slope is $$-5$$

Question 9

Which of the following is the correct derivative of $$y = \frac{3}{\sqrt{x}}$$?

Accepted Answer

$$y' = -\frac{3}{2}x$$^{-3/2}$$

Question 10

For the function $$y = x^2$ - 3$$, at what point is the tangent line horizontal?

Accepted Answer

At $$x = 0$$

Question 11

If $$y = x + \frac{2}{x^3$}$$, what is $$\frac{dy}{dx}$$ evaluated at $$x = 1$$?

Accepted Answer

$$\frac{dy}{dx}|_{x=1} = 1 - 6$$

Question 12

Which of the following is the correct derivative of $$y = -5x$$^{3/5}$$ - \frac{4}{\sqrt[3]{x}}$$?

Accepted Answer

$$y' = -3x$$^{2/5}$$ - \frac{4}{3}x$$^{-2/3}$$

Question 13

For $$f(x) = x^2$ - 9x + 1$$, what is $$f'(2)$$?

Accepted Answer

$$f'(2) = 4 - 9$$

Question 14

If $$y = 6x + \frac{2}{x^4$}$$, what is $$\frac{dy}{dx}$$ evaluated at $$x = 1$$?

Accepted Answer

$$\frac{dy}{dx}|_{x=1} = 6 - 8$$

Question 15

For $$f(x) = \frac{1}{x^2$}$$, what is the equation of the tangent line at $$(1, 1)$$?

Accepted Answer

$$y = -2x + 3$$

Question 16

Which of the following is the correct derivative of $$y = x^7$ - 2x^3$ + x^2$?

Accepted Answer

$$y' = 7x^6$ - 6x^2$ + 2x$$

Question 17

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

Accepted Answer

$$\frac{dy}{dx} = 2x + 1$$

Question 18

For the function $$w(x) = 0.00027744x^3$ - 0.022137x^2$ + 0.726618x + 3.51722$$, what is the approximate weight increase in week 2?

Accepted Answer

Approximately $$0.68 kg$$

Differentiation Techniques: Power, Sum, and Difference Rules in Business Calculus

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