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AP Calculus AB Quiz

AP Calculus AB Quiz: Applying The Power Rule

Practice Applying The Power Rule in AP Calculus AB with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 20

0 of 20 answered

The derivative of f(x)=4x5−6x32x2f(x) = \frac{4x^5 - 6x^3}{2x^2}f(x)=2x24x5−6x3​ for x≠0x \neq 0x=0 is

Select an answer to continue

What this quiz covers

This quiz focuses on Applying The Power Rule, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus AB.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The derivative of f(x)=4x5−6x32x2f(x) = \frac{4x^5 - 6x^3}{2x^2}f(x)=2x24x5−6x3​ for x≠0x \neq 0x=0 is

  1. 6x2−36x^2 - 36x2−3 (correct answer)
  2. 4x−34x - 34x−3
  3. 20x4−18x24x\frac{20x^4 - 18x^2}{4x}4x20x4−18x2​
  4. 23x4−32x2\frac{2}{3}x^4 - \frac{3}{2}x^232​x4−23​x2

Explanation: First, simplify the expression for f(x)f(x)f(x) by dividing each term in the numerator by the denominator: f(x)=4x52x2−6x32x2=2x3−3xf(x) = \frac{4x^5}{2x^2} - \frac{6x^3}{2x^2} = 2x^3 - 3xf(x)=2x24x5​−2x26x3​=2x3−3x. Now, differentiate the simplified polynomial using the power rule: f′(x)=2(3x2)−3(1)=6x2−3f'(x) = 2(3x^2) - 3(1) = 6x^2 - 3f′(x)=2(3x2)−3(1)=6x2−3.

Question 2

At which of the following values of xxx does the graph of f(x)=13x3−52x2+6x−1f(x) = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x - 1f(x)=31​x3−25​x2+6x−1 have a horizontal tangent line?

  1. x=1x=1x=1
  2. x=3x=3x=3 (correct answer)
  3. x=52x=\frac{5}{2}x=25​
  4. x=6x=6x=6

Explanation: A horizontal tangent line occurs where the derivative is equal to zero. First, find the derivative of f(x)f(x)f(x) using the power rule: f′(x)=x2−5x+6f'(x) = x^2 - 5x + 6f′(x)=x2−5x+6. Set the derivative equal to zero and solve for xxx: x2−5x+6=0x^2 - 5x + 6 = 0x2−5x+6=0. Factoring the quadratic gives (x−2)(x−3)=0(x-2)(x-3) = 0(x−2)(x−3)=0. The solutions are x=2x=2x=2 and x=3x=3x=3. Of the choices given, x=3x=3x=3 is an answer.

Question 3

If f(x)=5x4−3x2+2x−7f(x) = 5x^4 - 3x^2 + 2x - 7f(x)=5x4−3x2+2x−7, what is f′(x)f'(x)f′(x)?

  1. 20x3−6x+220x^3 - 6x + 220x3−6x+2 (correct answer)
  2. 20x4−6x2+220x^4 - 6x^2 + 220x4−6x2+2
  3. 5x3−3x+25x^3 - 3x + 25x3−3x+2
  4. x5−x3+x2−7xx^5 - x^3 + x^2 - 7xx5−x3+x2−7x

Explanation: The derivative of a polynomial is found by applying the power rule, d/dx(xn)=nxn−1d/dx(x^n) = nx^{n-1}d/dx(xn)=nxn−1, to each term. For f(x)=5x4−3x2+2x1−7x0f(x) = 5x^4 - 3x^2 + 2x^1 - 7x^0f(x)=5x4−3x2+2x1−7x0, the derivative is f′(x)=5(4x4−1)−3(2x2−1)+2(1x1−1)−7(0x0−1)=20x3−6x+2f'(x) = 5(4x^{4-1}) - 3(2x^{2-1}) + 2(1x^{1-1}) - 7(0x^{0-1}) = 20x^3 - 6x + 2f′(x)=5(4x4−1)−3(2x2−1)+2(1x1−1)−7(0x0−1)=20x3−6x+2.

Question 4

The function is f(x)=3x8−10x4+x3f(x)=3x^8-10x^4+x^3f(x)=3x8−10x4+x3. What is f′(x)f'(x)f′(x)?

  1. 24x7−40x3+3x224x^7-40x^3+3x^224x7−40x3+3x2 (correct answer)
  2. 24x8−40x4+3x324x^8-40x^4+3x^324x8−40x4+3x3
  3. 8x7−4x3+3x28x^7-4x^3+3x^28x7−4x3+3x2
  4. 24x7−10x3+3x224x^7-10x^3+3x^224x7−10x3+3x2
  5. 24x7−40x3+x224x^7-40x^3+x^224x7−40x3+x2

Explanation: This problem asks us to apply the power rule to find f'(x) where f(x) = 3x^8 - 10x^4 + x^3. The power rule states that the derivative of x^n is nx^(n-1). Applying this to each term: the derivative of 3x^8 is 8·3x^7 = 24x^7, the derivative of -10x^4 is 4·(-10)x^3 = -40x^3, and the derivative of x^3 is 3x^2. Thus, f'(x) = 24x^7 - 40x^3 + 3x^2. Choice E incorrectly gives the coefficient of x^2 as 1 instead of 3, forgetting to multiply by the original exponent. Remember that the power rule requires multiplying by the exponent, even when the original coefficient is 1.

Question 5

The cost function is C(x)=9x4−2x3+xC(x)=9x^4-2x^3+xC(x)=9x4−2x3+x. What is C′(x)C'(x)C′(x)?

  1. 36x4−6x3+136x^4-6x^3+136x4−6x3+1
  2. 36x3−6x2+136x^3-6x^2+136x3−6x2+1 (correct answer)
  3. 9x3−2x2+19x^3-2x^2+19x3−2x2+1
  4. 36x3−6x2+x36x^3-6x^2+x36x3−6x2+x
  5. 36x3−6x236x^3-6x^236x3−6x2

Explanation: This problem asks us to apply the power rule to find the derivative of a cost function. The power rule tells us that the derivative of x^n is nx^(n-1). For C(x) = 9x^4 - 2x^3 + x, we differentiate term by term: the derivative of 9x^4 is 4·9x^3 = 36x^3, the derivative of -2x^3 is 3·(-2)x^2 = -6x^2, and the derivative of x (which is x^1) is 1·x^0 = 1. Thus, C'(x) = 36x^3 - 6x^2 + 1. Choice E incorrectly omits the derivative of the x term, forgetting that x has a derivative of 1. Always include the derivatives of all terms, even when the coefficient is 1.

Question 6

The function m(x)=x10−4x7+8xm(x)=x^{10}-4x^7+8xm(x)=x10−4x7+8x models a quantity. What is m′(x)m'(x)m′(x)?

  1. 10x9−28x6+810x^9-28x^6+810x9−28x6+8 (correct answer)
  2. 10x10−28x7+8x10x^{10}-28x^7+8x10x10−28x7+8x
  3. 9x9−6x6+89x^9-6x^6+89x9−6x6+8
  4. 10x9−4x6+810x^9-4x^6+810x9−4x6+8
  5. 10x9−28x610x^9-28x^610x9−28x6

Explanation: This problem requires applying the power rule to differentiate m(x) = x^10 - 4x^7 + 8x. The power rule states that d/dx[x^n] = nx^(n-1). Applying this to each term: the derivative of x^10 is 10x^9, the derivative of -4x^7 is 7·(-4)x^6 = -28x^6, and the derivative of 8x is 8·1x^0 = 8. Thus, m'(x) = 10x^9 - 28x^6 + 8. Choice E incorrectly omits the derivative of the 8x term, forgetting that linear terms have constant derivatives. Always differentiate every term in the function, including linear terms whose derivatives are constants.

Question 7

A cost function is C(x)=7x2−4x5+10C(x)=7x^2-4x^5+10C(x)=7x2−4x5+10. What is C′(x)C'(x)C′(x)?

  1. 14x−20x414x-20x^414x−20x4 (correct answer)
  2. 14x2−20x514x^2-20x^514x2−20x5
  3. 7x−4x47x-4x^47x−4x4
  4. 14x−20x4+1014x-20x^4+1014x−20x4+10
  5. 2x−20x42x-20x^42x−20x4

Explanation: This problem requires applying the power rule to find the derivative of a cost function that is a polynomial. Per the power rule, the derivative of c x^n is c n x^{n-1}, applied term by term. For 7x^2, it is 14x; for -4x^5, it becomes -20x^4; and +10 differentiates to 0. Thus, C'(x) = 14x - 20x^4. A tempting distractor is choice D, which wrongly retains the constant 10, but constants differentiate to zero. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 8

A simple model is p(t)=t6+4t2−11p(t)=t^6+4t^2-11p(t)=t6+4t2−11. What is p′(t)p'(t)p′(t)?

  1. 6t5+8t6t^5+8t6t5+8t (correct answer)
  2. 6t6+8t26t^6+8t^26t6+8t2
  3. t5+4tt^5+4tt5+4t
  4. 6t5+8t−116t^5+8t-116t5+8t−11
  5. 5t5+8t5t^5+8t5t5+8t

Explanation: This problem requires applying the power rule to find the derivative of a simple polynomial model. The power rule dictates that for c t^n, the derivative is c n t^{n-1}, handling each term separately. Differentiating t^6 (1t^6) gives 6t^5, +4t^2 becomes +8t, and -11 turns to 0. Combining yields p'(t) = 6t^5 + 8t. A tempting distractor is choice D, which incorrectly includes the constant -11 in the derivative. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 9

The revenue is modeled by R(x)=3x8−2x4+xR(x)=3x^8-2x^4+xR(x)=3x8−2x4+x. Find R′(x)R'(x)R′(x).

  1. 24x7−8x3+124x^7-8x^3+124x7−8x3+1 (correct answer)
  2. 24x8−8x4+124x^8-8x^4+124x8−8x4+1
  3. 3x7−2x3+13x^7-2x^3+13x7−2x3+1
  4. 24x7−8x3+x24x^7-8x^3+x24x7−8x3+x
  5. 8x7−8x3+18x^7-8x^3+18x7−8x3+1

Explanation: This problem requires applying the power rule to find the derivative of a revenue model polynomial. Per the power rule, the derivative of c x^n is c n x^{n-1}, applied to each term. For 3x^8, it becomes 24x^7; for -2x^4, it is -8x^3; and +x (x^1) differentiates to +1. Thus, R'(x) = 24x^7 - 8x^3 + 1. A tempting distractor is choice D, which wrongly changes the constant 1 to x. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 10

A particle’s position is modeled by s(t)=4t5−3t2+7s(t)=4t^5-3t^2+7s(t)=4t5−3t2+7. What is s′(t)s'(t)s′(t)?

  1. 20t4−6t20t^4-6t20t4−6t (correct answer)
  2. 4t4−3t+74t^4-3t+74t4−3t+7
  3. 20t5−6t220t^5-6t^220t5−6t2
  4. 5t4−2t5t^4-2t5t4−2t
  5. 20t4−6t+720t^4-6t+720t4−6t+7

Explanation: This problem requires applying the power rule to find the derivative of a position function. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). For s(t) = 4t^5 - 3t^2 + 7, we differentiate each term: the derivative of 4t^5 is 5·4t^4 = 20t^4, the derivative of -3t^2 is 2·(-3)t^1 = -6t, and the derivative of the constant 7 is 0. Therefore, s'(t) = 20t^4 - 6t. Choice E incorrectly keeps the constant term 7, failing to recognize that constants have zero derivatives. When differentiating polynomials, apply the power rule term by term and remember that constants disappear.

Question 11

A height function is H(t)=15t4+3t3−2H(t)=15t^4+3t^3-2H(t)=15t4+3t3−2. What is H′(t)H'(t)H′(t)?

  1. 60t3+9t260t^3+9t^260t3+9t2 (correct answer)
  2. 60t4+9t3−260t^4+9t^3-260t4+9t3−2
  3. 60t3+3t260t^3+3t^260t3+3t2
  4. 15t3+3t215t^3+3t^215t3+3t2
  5. 60t3+9t2−260t^3+9t^2-260t3+9t2−2

Explanation: This problem asks us to apply the power rule to differentiate a height function. The power rule states that the derivative of t^n is nt^(n-1). For H(t) = 15t^4 + 3t^3 - 2, we differentiate term by term: the derivative of 15t^4 is 4·15t^3 = 60t^3, the derivative of 3t^3 is 3·3t^2 = 9t^2, and the derivative of the constant -2 is 0. Therefore, H'(t) = 60t^3 + 9t^2. Choice E incorrectly retains the constant -2, not recognizing that constants have zero derivatives. When differentiating, always remember that constant terms vanish completely.

Question 12

A tank’s volume is V(t)=t6−8t3+12V(t)=t^6-8t^3+12V(t)=t6−8t3+12. What is V′(t)V'(t)V′(t)?

  1. 6t5−24t26t^5-24t^26t5−24t2 (correct answer)
  2. 6t6−24t36t^6-24t^36t6−24t3
  3. t5−8t2t^5-8t^2t5−8t2
  4. 6t5−24t2+126t^5-24t^2+126t5−24t2+12
  5. 6t5−8t26t^5-8t^26t5−8t2

Explanation: This problem requires applying the power rule to differentiate a volume function. The power rule states that d/dx[x^n] = nx^(n-1). For V(t) = t^6 - 8t^3 + 12, we apply the rule to each term: the derivative of t^6 is 6t^5, the derivative of -8t^3 is 3·(-8)t^2 = -24t^2, and the derivative of the constant 12 is 0. Therefore, V'(t) = 6t^5 - 24t^2. Choice D incorrectly retains the constant 12, not recognizing that constants have zero derivatives. Remember that when differentiating, constant terms always disappear, leaving only the derivatives of the variable terms.

Question 13

A population model is P(t)=2t9+6t2−4tP(t)=2t^9+6t^2-4tP(t)=2t9+6t2−4t. What is P′(t)P'(t)P′(t)?

  1. 18t8+12t−418t^8+12t-418t8+12t−4 (correct answer)
  2. 18t8+12t−4t18t^8+12t-4t18t8+12t−4t
  3. 18t9+12t2−418t^9+12t^2-418t9+12t2−4
  4. 18t8+12t2−418t^8+12t^2-418t8+12t2−4
  5. 2t8+12t−42t^8+12t-42t8+12t−4

Explanation: This problem involves applying the power rule to differentiate a population model. The power rule states that if f(t) = t^n, then f'(t) = nt^(n-1). For P(t) = 2t^9 + 6t^2 - 4t, we differentiate each term: the derivative of 2t^9 is 9·2t^8 = 18t^8, the derivative of 6t^2 is 2·6t^1 = 12t, and the derivative of -4t is -4·1t^0 = -4. Therefore, P'(t) = 18t^8 + 12t - 4. Choice D incorrectly writes 12t^2 instead of 12t, adding an extra power to the middle term. When applying the power rule, always reduce the exponent by exactly one.

Question 14

A model gives h(x)=12x3−5x5+2x2h(x)=12x^3-5x^5+2x^2h(x)=12x3−5x5+2x2. What is h′(x)h'(x)h′(x)?

  1. 36x2−25x4+4x36x^2-25x^4+4x36x2−25x4+4x (correct answer)
  2. 36x3−25x5+4x236x^3-25x^5+4x^236x3−25x5+4x2
  3. 36x2−5x4+4x36x^2-5x^4+4x36x2−5x4+4x
  4. 12x2−25x4+2x12x^2-25x^4+2x12x2−25x4+2x
  5. 36x2−25x5+4x36x^2-25x^5+4x36x2−25x5+4x

Explanation: This problem requires applying the power rule to differentiate a polynomial function. The power rule tells us that d/dx[x^n] = nx^(n-1). For h(x) = 12x^3 - 5x^5 + 2x^2, we differentiate term by term: the derivative of 12x^3 is 3·12x^2 = 36x^2, the derivative of -5x^5 is 5·(-5)x^4 = -25x^4, and the derivative of 2x^2 is 2·2x^1 = 4x. Therefore, h'(x) = 36x^2 - 25x^4 + 4x. Choice C incorrectly computes the coefficient of x^4 as -5 instead of -25, failing to multiply properly. Always multiply the original coefficient by the exponent when applying the power rule.

Question 15

A function is q(x)=8x2+5x7q(x)=8x^2+5x^7q(x)=8x2+5x7. Find q′(x)q'(x)q′(x).

  1. 16x+35x616x+35x^616x+35x6 (correct answer)
  2. 16x2+35x716x^2+35x^716x2+35x7
  3. 8x+5x68x+5x^68x+5x6
  4. 16x+35x716x+35x^716x+35x7
  5. 10x+35x610x+35x^610x+35x6

Explanation: This problem requires applying the power rule to find the derivative of the polynomial q(x). The power rule states that for c x^n, the derivative is c n x^{n-1}, treating terms separately. Differentiating 8x^2 gives 16x, and +5x^7 becomes +35x^6. Combining these yields q'(x) = 16x + 35x^6. A tempting distractor is choice D, which incorrectly uses 35x^7 instead of 35x^6. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 16

A polynomial model is f(x)=10x4−7x6+3x2f(x)=10x^4-7x^6+3x^2f(x)=10x4−7x6+3x2. What is f′(x)f'(x)f′(x)?

  1. 40x3−42x5+6x40x^3-42x^5+6x40x3−42x5+6x (correct answer)
  2. 40x4−42x6+6x240x^4-42x^6+6x^240x4−42x6+6x2
  3. 10x3−7x5+3x10x^3-7x^5+3x10x3−7x5+3x
  4. 40x3−7x5+6x40x^3-7x^5+6x40x3−7x5+6x
  5. 40x3−42x5+3x240x^3-42x^5+3x^240x3−42x5+3x2

Explanation: This problem involves applying the power rule to find the derivative of a polynomial function. The power rule states that if f(x) = ax^n, then f'(x) = n·ax^(n-1). For f(x) = 10x^4 - 7x^6 + 3x^2, we apply the rule to each term: the derivative of 10x^4 is 10·4x^3 = 40x^3, the derivative of -7x^6 is -7·6x^5 = -42x^5, and the derivative of 3x^2 is 3·2x^1 = 6x. Thus, f'(x) = 40x^3 - 42x^5 + 6x. Choice C shows the error of not multiplying coefficients by powers, giving 10x^3 - 7x^5 + 3x. Remember: the power rule requires multiplying the original coefficient by the exponent before decreasing the exponent.

Question 17

A polynomial model is f(x)=x7−3x3+2x2f(x)=x^7-3x^3+2x^2f(x)=x7−3x3+2x2. Find f′(x)f'(x)f′(x).

  1. 7x6−9x2+4x7x^6-9x^2+4x7x6−9x2+4x (correct answer)
  2. 7x7−9x3+4x27x^7-9x^3+4x^27x7−9x3+4x2
  3. x6−3x2+2xx^6-3x^2+2xx6−3x2+2x
  4. 7x6−9x2+4x27x^6-9x^2+4x^27x6−9x2+4x2
  5. 6x6−9x2+4x6x^6-9x^2+4x6x6−9x2+4x

Explanation: This problem requires applying the power rule to find the derivative of a given polynomial model. The power rule indicates that for c x^n, the derivative is c n x^{n-1}, handling each term on its own. Differentiating x^7 (which is 1x^7) gives 7x^6, -3x^3 becomes -9x^2, and +2x^2 turns into +4x. Combining yields f'(x) = 7x^6 - 9x^2 + 4x. A tempting distractor is choice D, which incorrectly uses 4x^2 instead of 4x for the last term. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 18

A function is g(x)=12x3−6x2+4g(x)=12x^3-6x^2+4g(x)=12x3−6x2+4. What is g′(x)g'(x)g′(x)?

  1. 36x2−12x36x^2-12x36x2−12x (correct answer)
  2. 36x3−12x236x^3-12x^236x3−12x2
  3. 12x2−6x12x^2-6x12x2−6x
  4. 36x2−12x+436x^2-12x+436x2−12x+4
  5. 24x2−12x24x^2-12x24x2−12x

Explanation: This problem requires applying the power rule to find the derivative of the given polynomial function g(x). The power rule states that for c x^n, the derivative is c n x^{n-1}, treating terms individually. Differentiating 12x^3 gives 36x^2, -6x^2 becomes -12x, and +4 turns to 0. Combining these yields g'(x) = 36x^2 - 12x. A tempting distractor is choice D, which incorrectly includes the constant 4 in the derivative. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 19

A polynomial model is y(x)=6x6+5x5−4x3y(x)=6x^6+5x^5-4x^3y(x)=6x6+5x5−4x3. Find y′(x)y'(x)y′(x).

  1. 36x5+25x4−12x236x^5+25x^4-12x^236x5+25x4−12x2 (correct answer)
  2. 36x6+25x5−12x336x^6+25x^5-12x^336x6+25x5−12x3
  3. 6x5+5x4−4x26x^5+5x^4-4x^26x5+5x4−4x2
  4. 36x5+25x4−12x336x^5+25x^4-12x^336x5+25x4−12x3
  5. 30x5+25x4−12x230x^5+25x^4-12x^230x5+25x4−12x2

Explanation: This problem requires applying the power rule to find the derivative of a polynomial model y(x). According to the power rule, the derivative of c x^n is c n x^{n-1}, applied term by term. For 6x^6, it becomes 36x^5; for +5x^5, it is +25x^4; and -4x^3 differentiates to -12x^2. Thus, y'(x) = 36x^5 + 25x^4 - 12x^2. A tempting distractor is choice D, which wrongly uses -12x^3 instead of -12x^2. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.

Question 20

Let f(x)=10x5−3x4−2xf(x)=10x^5-3x^4-2xf(x)=10x5−3x4−2x. What is the derivative f′(x)f'(x)f′(x)?

  1. 50x4−12x3−250x^4-12x^3-250x4−12x3−2 (correct answer)
  2. 50x5−12x4−2x50x^5-12x^4-2x50x5−12x4−2x
  3. 10x4−3x3−210x^4-3x^3-210x4−3x3−2
  4. 50x4−12x3−2x50x^4-12x^3-2x50x4−12x3−2x
  5. 40x4−12x3−240x^4-12x^3-240x4−12x3−2

Explanation: This problem requires applying the power rule to find the derivative of the polynomial f(x). According to the power rule, the derivative of c x^n is c n x^{n-1}, applied term by term. For 10x^5, it becomes 50x^4; for -3x^4, it is -12x^3; and -2x differentiates to -2. Thus, f'(x) = 50x^4 - 12x^3 - 2. A tempting distractor is choice D, which wrongly changes -2 to -2x. When differentiating polynomials, apply the power rule independently to each term and combine the results, remembering constants become zero.