Question 1

Which rule allows you to differentiate a function multiplied by a constant, such as $$c \cdot f(x)$$?

Accepted Answer

Constant Multiple Rule

Question 2

What is the derivative of $$f(x) = 7x$$?

Accepted Answer

$$7$$

Question 3

Find the derivative of $$f(x) = -5x$$^{20}$$.

Accepted Answer

$$-100x^{19}$$

Question 4

If $$f(x) = 6x^2$ - 8x$$, at which point does the slope of the tangent line equal $$16$$?

Accepted Answer

$$(2, 8)$$

Question 5

For the function $$f(x) = x^3$ + 4x^2$ - 3x$$, at which $$x-$$values is the tangent line horizontal?

Accepted Answer

$$x = -3$$ and $$x = \frac{1}{3}$$

Question 6

Find the equation of the tangent line to $$f(x) = x^3$ + 4x^2$ - 3x at $$the point $$(1, 2).$$

Accepted Answer

$$y = 8x - 6$$

Question 7

What is the derivative of $$f(x) = 6\sqrt{x}$$?

Accepted Answer

$$3x^{-1/2}$$

Question 8

Given $$f(x) = 12x^5$ - 40x + e^x - e^4$, what is f$$'(x)$$?

Accepted Answer

$$60x^4 - 40 + e^x$

Question 9

What is the derivative of $$f(x) = e^x$$?

Accepted Answer

$$e^x$$

Question 10

Find the first and second derivatives of $$f(x) = 2x^3$ + 9x^2$ + 6x.$$

Accepted Answer

First: $$6x^2 + 18x + 6$, Second: 12x + 18$

Question 11

Which rule is used to differentiate $$f(x) = x^n$$?

Accepted Answer

Power Rule

Question 12

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

Accepted Answer

$$-15x^{-4}$$ - $$4x^{-2}$$

Question 13

A chemical reaction rate is modeled by $$R(t) = 12t^5$ - 40t + e^t - e^4$. What is the instantaneous rate of change at t=0$?

Accepted Answer

$$-40 + 1$$

Question 14

Which of the following is the correct expression for the $$nth $$derivative of $$f(x)$$?

Accepted Answer

$$f^{(n)}(x$$) = $$(f^{(n-1)}(x$$))'$$

Question 15

A medical dosage is modeled by $$D(x) = 6x^2$ - 8x. $$What is the rate of change of dosage at $$x = 3$$?

Accepted Answer

$$28$$

Question 16

Which rule allows you to differentiate $$f(x) + g(x)$$?

Accepted Answer

Sum/Difference Rule

Question 17

If $$f(x) = 2x^3$ + 9x^2$ + 6x$$, what is $$f''(x)$$?

Accepted Answer

$$12x + 18$$

Question 18

A population grows according to $$P(t) = e^t. $$What is the instantaneous rate of growth at any time $$t$$?

Accepted Answer

$$e^t$$

Question 19

Which of the following is the correct formula for the second derivative of $$f(x)$$?

Accepted Answer

$$f''(x) = (f'(x))'$$

Rules of Differentiation: Fundamental Techniques and Applications 3.3

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