AP Calculus AB Practice Quiz: Applying the Power Rule

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

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

A)7x^6B)x^6C)7x^8D)x^8 / 8

All Questions (7)

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

A) 7x^6

B) x^6

C) 7x^8

D) x^8 / 8

Correct Answer: A

According to the power rule for derivatives, for a function of the form $f(x)=x^r$, the derivative is $f'(x)=rx^{r-1}$. For $f(x) = x^7$, we have $r=7$. Applying the rule, the derivative is $f'(x) = 7x^{7-1} = 7x^6$.

If $g(x) = 5x^4$, what is $g'(x)$?

A) 20x^3

B) 5x^3

C) 20x^5

D) 4x^3

Correct Answer: A

To find the derivative of $g(x) = 5x^4$, we use the constant multiple rule and the power rule. The power rule states that the derivative of $x^r$ is $rx^{r-1}$. The constant multiple rule states that the derivative of $c \\cdot f(x)$ is $c \\cdot f'(x)$. Therefore, $g'(x) = 5 \\cdot \\frac{d}{dx}(x^4) = 5 \\cdot (4x^{4-1}) = 20x^3$.

Find the derivative of the function $h(x) = \\sqrt[3]{x}$.

A) \\frac{1}{3}x^{-2/3}

B) 3x^{2/3}

C) \\frac{1}{3}x^{1/3}

D) x^{-2/3}

Correct Answer: A

To apply the power rule, first rewrite the function using a rational exponent: $h(x) = \\sqrt[3]{x} = x^{1/3}$. Now, apply the power rule $\\frac{d}{dx}(x^r) = rx^{r-1}$ with $r = 1/3$. The derivative is $h'(x) = \\frac{1}{3}x^{(1/3) - 1} = \\frac{1}{3}x^{-2/3}$.

What is the derivative of $f(x) = \\frac{2}{x^3}$?

A) -6x^{-4}

B) 2x^{-2}

C) -6x^{-2}

D) 6x^{-4}

Correct Answer: A

First, rewrite the function using a negative exponent to apply the power rule: $f(x) = 2x^{-3}$. Now, use the constant multiple rule and the power rule, where $r = -3$. The derivative is $f'(x) = 2 \\cdot \\frac{d}{dx}(x^{-3}) = 2 \\cdot (-3x^{-3-1}) = -6x^{-4}$.

Let $f(x) = 3x^2 - 8x + 1$. What is $f'(x)$?

A) 6x - 8

B) 3x - 8

C) 6x + 1

D) x^3 - 4x^2 + x

Correct Answer: A

To find the derivative of the polynomial, we apply the power rule to each term separately. The derivative of $3x^2$ is $3 \\cdot 2x^{2-1} = 6x$. The derivative of $-8x$ (or $-8x^1$) is $-8 \\cdot 1x^{1-1} = -8x^0 = -8$. The derivative of a constant, 1, is 0. Combining these results, $f'(x) = 6x - 8 + 0 = 6x - 8$.

Find the derivative, $\\frac{dy}{dx}$, for the function $y = 4\\sqrt{x} + \\frac{1}{\\sqrt{x}}$.

A) 2x^{-1/2} - \\frac{1}{2}x^{-3/2}

B) 4x^{1/2} + x^{-1/2}

C) 2x^{1/2} - \\frac{1}{2}x^{-1/2}

D) 8x^{-1/2} + \\frac{1}{2}x^{-3/2}

Correct Answer: A

First, rewrite the function using rational exponents: $y = 4x^{1/2} + x^{-1/2}$. Now, apply the power rule to each term. For the first term, $\\frac{d}{dx}(4x^{1/2}) = 4 \\cdot \\frac{1}{2}x^{(1/2)-1} = 2x^{-1/2}$. For the second term, $\\frac{d}{dx}(x^{-1/2}) = -\\frac{1}{2}x^{(-1/2)-1} = -\\frac{1}{2}x^{-3/2}$. Combining the terms, $\\frac{dy}{dx} = 2x^{-1/2} - \\frac{1}{2}x^{-3/2}$.

If $f(x) = \\frac{x^3 - 2x}{x}$, what is the value of $f'(2)$?

A) 4

B) 2

C) 3

D) 5

Correct Answer: A

First, simplify the function $f(x)$ before differentiating. $f(x) = \\frac{x^3}{x} - \\frac{2x}{x} = x^2 - 2$ (for $x \\neq 0$). Now, find the derivative using the power rule: $f'(x) = \\frac{d}{dx}(x^2) - \\frac{d}{dx}(2) = 2x - 0 = 2x$. Finally, evaluate the derivative at $x=2$: $f'(2) = 2(2) = 4$.