Question 1

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

Accepted Answer

$$\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$$

Question 2

Given $$f(x) = x^2$ + 1$$ and $$g(x) = x^2$ - 4$$, what is the derivative of $$\frac{f(x)}{g(x)}$$ using the Quotient Rule?

Accepted Answer

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

Question 3

What is the slope of the tangent line to $$f(x) = \frac{x^2$ + 1}{x^2$ - 4} at $$the point $$(3, 2)$$?

Accepted Answer

$$-\frac{6}{5}$$

Question 4

Which of the following is the equation of the tangent line to $$f(x) = \frac{x^2$ + 1}{x^2$ - 4} at$$ $$(3, 2)$$?

Accepted Answer

$$y = -\frac{6}{5}x + \frac{28}{5}$$

Question 5

If $$n is $$any real number, what is the general form of the Power Rule for differentiation?

Accepted Answer

$$\frac{d}{dx}(x^n) = nx$$^{n-1}$$

Question 6

Using the Power Rule, what is $$\frac{d}{dx}(3x$$^{10}$$ - 4x$$^{-6}$$)$$?

Accepted Answer

$$30x^9$$ + $$24x^{-7}$$

Question 7

Which of the following is the correct application of the Quotient Rule for $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$$?

Accepted Answer

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

Question 8

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = \frac{4e^t}{t^2$ + 1}. $$What is $$\frac{dC}{dt}$$?

Accepted Answer

$$\frac{4(t^2$ + 1)e^t - 4e^t(2t)}{(t^2$ + 1)^2$}$$

Question 9

Which of the following is the derivative of $$y = x^2$ e^x$$ using the Product Rule?

Accepted Answer

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

Question 10

A physicist models the velocity of a particle as $$v(t) = \frac{t^3$ + 3t + 4}{t^2$ - 1}. $$What is $$v'(t)$$?

Accepted Answer

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

Question 11

If $$f(x) = x^3$ and g(x) = e^x$, what is $$\frac{d}{dx}(f(x)g(x))$$?

Accepted Answer

$$3x^2 e^x$$ + $$x^3 e^x$

Question 12

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

Accepted Answer

$$-e^{-2x}$$

Question 13

A biologist models population growth as $$P(t) = t^2$ \sqrt{t}. $$What is $$\frac{dP}{dt}$$?

Accepted Answer

$$\frac{5}{2} t$$^{3/2}$$

Question 14

Which of the following is the correct derivative of $$y = \frac{x^2$ + 1}{x^2$ + 1}$$?

Accepted Answer

$$0$$

Question 15

If $$f(x) = x^2$ and g(x) = 2x$, what is $$\frac{d}{dx}(f(x)g(x))$$?

Accepted Answer

Question 16

Which of the following is the correct derivative of $$y = \sqrt{2V} + 1$$ with respect to $$V$$?

Accepted Answer

$$\frac{1}{\sqrt{2V}}$$

Product and Quotient Rules in Differentiation

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