Question 1

Given the function $h(x) = (x^3 - 3)(x^4 + 8)$, what is $h'(x)$?

Accepted Answer

$h'(x) = 3x^2(x^4 + 8) + 4x^3(x^3 - 3)$

Question 2

Which of the following correctly applies the Quotient Rule to $f(x) = \dfrac{x^2 - 4}{x + 2}$?

Accepted Answer

$f'(x) = \dfrac{(x + 2)2x - (x^2 - 4) \cdot 1}{(x + 2)^2}$

Question 3

Suppose $f(x) = x^3 - x^2 - 12x + 1$. For which $x$ does $f(x)$ have a horizontal tangent line?

Accepted Answer

$x = 2$ and $x = -2$

Question 4

If $f(x) = \dfrac{1}{\sqrt{x}}$, what is $f'(x)$?

Accepted Answer

$f'(x) = -\dfrac{1}{2x^{3/2}}$

Question 5

Given $g(x) = x^3 f(x)$ and $f(2) = 2$, $f'(2) = 3$, what is $g'(2)$?

Accepted Answer

$g'(2) = 3 \cdot 4 \cdot 2 + 8 \cdot 3 = 24 + 24 = 48$

Question 6

Which of the following is the correct derivative of $y = \dfrac{5}{14x^7}$?

Accepted Answer

$y' = -\dfrac{5}{14} \cdot 7x^{-8}$

Question 7

If $h(x) = \dfrac{f(x)}{x-3}$, and $f(2) = 2$, $f'(2) = 3$, what is $h'(2)$?

Accepted Answer

$h'(2) = \dfrac{f'(2)(2-3) - f(2) \cdot 1}{(2-3)^2} = \dfrac{3(-1) - 2}{1} = -3 - 2 = -5$

Question 8

Which of the following statements about the derivative of a constant function $f(x) = c$ is true?

Accepted Answer

$f'(x) = 0$

Question 9

Let $f(x) = x^n$ where $n$ is a rational number. What is $f'(x)$?

Accepted Answer

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

Question 10

Given $y = \dfrac{5}{(2x)^4}$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{dy}{dx} = -\dfrac{5}{8x^5}$

Question 11

If $f(x) = 3x^2 - 7x^{1/2}$, what is $f''(x)$?

Accepted Answer

$f''(x) = 6 - \dfrac{7}{4}x^{-3/2}$

Question 12

A function $f(x)$ has a horizontal tangent at $x = a$ if:

Accepted Answer

$f'(a) = 0$

Question 13

Given $h(t) = \dfrac{1}{t^{1/3}}$, what is $h'(t)$?

Accepted Answer

$h'(t) = -\dfrac{1}{3} t^{-4/3}$

Question 14

If $f(x) = \sqrt[3]{x}$, what is $f'(x)$?

Accepted Answer

$f'(x) = \dfrac{1}{3} x^{-2/3}$

Question 15

Which of the following is the correct application of the Product Rule for $\dfrac{d}{dx}[f(x)g(x)]$?

Accepted Answer

$f'(x)g(x) + f(x)g'(x)$

Question 16

Suppose $f(x) = x^2 - 4$ and $g(x) = x + 2$. Using the Quotient Rule, what is $\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)$?

Accepted Answer

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

Question 17

If $f(x) = 6x^{1/2}$, what is $f'(x)$?

Accepted Answer

$f'(x) = 3x^{-1/2}$

Question 18

A real-world application: The position of a car is given by $s(t) = 3t^2 - 7t^{1/2}$. What is the car's acceleration at $t = 4$ seconds?

Accepted Answer

$a(4) = 6 - \dfrac{7}{4} \cdot 4^{-3/2} = 6 - \dfrac{7}{4} \cdot \dfrac{1}{8} = 6 - \dfrac{7}{32} = \dfrac{185}{32}$

Question 19

If $f(x) = x^n$ and $n$ is a rational number, what is the $k$th derivative $f^{(k)}(x)$?

Accepted Answer

$f^{(k)}(x) = n(n-1)\cdots(n-k+1)x^{n-k}$

Derivatives: Product Rule, Quotient Rule, and Power Rule

Study Guide - Practice Questions