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 $n$th 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

Study Guide - Practice Questions