Q: A business's revenue is $$R(t) = N(t)P(t)$$, where $$N(t) is $$the number of sales and $$P(t) is $$the price. If $$N(5) = 100$$, $$N'(5) = 3$$, $$P(5) = 18$$, and $$P'(5) = -1$$, what is $$R'(5)$$?

$$R'(5) = N'(5)P(5) + N(5)P'(5) = 3 \times 18 + 100 \times (-1) = 54 - 100 = -46$$

Question 1

Which of the following is the correct application of the power rule to find the derivative of $$f(x) = x^4$?

Accepted Answer

$$f'(x) = 4x^3$

Question 2

Given $$f(x) = 3x^2$ + 5e^x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x + 5e^x$$

Question 3

If $$h(x) = x e^x$$, what is $$h'(x)$$?

Accepted Answer

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

Question 4

Which rule is used to differentiate $$y = x^2$ e^x$$?

Accepted Answer

Product Rule

Question 5

Given $$y = \frac{3x-5}{1+2x^2$}$$, what is the derivative $$\frac{dy}{dx}$$ using the quotient rule?

Accepted Answer

$$\frac{(1+2x^2$)(3) - (3x-5)(4x)}{(1+2x^2$)^2$$}$

Question 6

A business's revenue is $$R(t) = N(t)P(t)$$, where $$N(t) is $$the number of sales and $$P(t) is $$the price. If $$N(5) = 100$$, $$N'(5) = 3$$, $$P(5) = 18$$, and $$P'(5) = -1$$, what is $$R'(5)$$?

Accepted Answer

$$R'(5) = N'(5)P(5) + N(5)P'(5) = 3 	imes 18 + 100 	imes (-1) = 54 - 100 = -46$$

Question 7

Find the derivative of $$g(x) = \frac{x^3$ - 2}{x}.$$

Accepted Answer

$$g'(x) = 2x + \frac{2}{x^2$}$$

Question 8

Which of the following best describes the meaning of the derivative $$f'(x) in a $$real-world context?

Accepted Answer

It represents the instantaneous rate of change of $$f(x)$$ with respect to $$x.$$

Question 9

Given $$y = \frac{x^2$ e^x}{1 + 3x^4$}$$, which rule(s) must be used to differentiate $$y$$?

Accepted Answer

Product Rule and Quotient Rule

Question 10

If $$g(x) = \frac{2x}{x^2$ + 1}$$, what is $$g'(2)$$?

Accepted Answer

$$g'(2) = \frac{-6}{25}$$

Question 11

A car's position is given by $$s(t) = t^2$ e^t. $$What is the car's velocity at time $$t$$?

Accepted Answer

$$v(t) = 2t e^t + t^2$ e^t$$

Question 12

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

Accepted Answer

$$\frac{d}{dx} \left( \frac{H}{L} \right) = \frac{L H' - H L'}{L^2$}$$

Question 13

If $$y = 4 \sin x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$4 \cos x$$

Question 14

Given $$f(x) = x^2$ + 3x + 5$$, what is the equation of the tangent line at $$x = 1$$?

Accepted Answer

$$y = 5x + 3$$

Question 15

Which of the following best describes the process for finding the slope of the tangent line to $$y = f(x) at$$ $$x = a$$?

Accepted Answer

Compute $$f'(a)$$

Techniques of Differentiation: Product and Quotient Rules

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