AP Calculus AB Quiz: Applying The Power Rule
20 questions · exam conditions
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Applying The Power RuleQuestion 1 of 20

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

24x7−40x3+3x224x^7-40x^3+3x^2
24x8−40x4+3x324x^8-40x^4+3x^3
8x7−4x3+3x28x^7-4x^3+3x^2
24x7−10x3+3x224x^7-10x^3+3x^2
24x7−40x3+x224x^7-40x^3+x^2
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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.

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

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

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 2

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

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

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 x1x^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 3

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

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

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 4

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

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

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

Question 5

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

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

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 6

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

  1. 6t5+8t6t^5+8t (correct answer)
  2. 6t6+8t26t^6+8t^2
  3. t5+4tt^5+4t
  4. 6t5+8t−116t^5+8t-11
  5. 5t5+8t5t^5+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 (1t61t^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 7

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

  1. 24x7−8x3+124x^7-8x^3+1 (correct answer)
  2. 24x8−8x4+124x^8-8x^4+1
  3. 3x7−2x3+13x^7-2x^3+1
  4. 24x7−8x3+x24x^7-8x^3+x
  5. 8x7−8x3+18x^7-8x^3+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 (x1x^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 8

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

  1. 20t4−6t20t^4-6t (correct answer)
  2. 4t4−3t+74t^4-3t+7
  3. 20t5−6t220t^5-6t^2
  4. 5t4−2t5t^4-2t
  5. 20t4−6t+720t^4-6t+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 9

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

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

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 10

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

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

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 11

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

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

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

  1. 36x2−25x4+4x36x^2-25x^4+4x (correct answer)
  2. 36x3−25x5+4x236x^3-25x^5+4x^2
  3. 36x2−5x4+4x36x^2-5x^4+4x
  4. 12x2−25x4+2x12x^2-25x^4+2x
  5. 36x2−25x5+4x36x^2-25x^5+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 13

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

  1. 16x+35x616x+35x^6 (correct answer)
  2. 16x2+35x716x^2+35x^7
  3. 8x+5x68x+5x^6
  4. 16x+35x716x+35x^7
  5. 10x+35x610x+35x^6

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 14

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

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

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 15

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

  1. 7x6−9x2+4x7x^6-9x^2+4x (correct answer)
  2. 7x7−9x3+4x27x^7-9x^3+4x^2
  3. x6−3x2+2xx^6-3x^2+2x
  4. 7x6−9x2+4x27x^6-9x^2+4x^2
  5. 6x6−9x2+4x6x^6-9x^2+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 1x71x^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 16

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

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

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 17

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

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

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 18

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

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

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.

Question 19

A company's cost is C(x)=7x5−4x2+9C(x)=7x^5-4x^2+9. What is C′(x)C'(x)?

  1. 35x4−8x35x^4-8x (correct answer)
  2. 35x5−8x235x^5-8x^2
  3. 7x4−4x+07x^4-4x+0
  4. 35x4−4x35x^4-4x
  5. 35x4−8x+935x^4-8x+9

Explanation: This problem involves applying 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) = 7x^5 - 4x^2 + 9, we differentiate each term separately: the derivative of 7x^5 is 7·5x^4 = 35x^4, the derivative of -4x^2 is -4·2x^1 = -8x, and the derivative of the constant 9 is 0. Thus, C'(x) = 35x^4 - 8x + 0 = 35x^4 - 8x. Choice C incorrectly keeps the +0 term and uses wrong coefficients, failing to multiply the original coefficients by the powers. Remember that constants always have a derivative of zero and can be omitted from the final answer.

Question 20

A model for distance is d(t)=2t8−3t2+5t4d(t)=2t^8-3t^2+5t^4. What is d′(t)d'(t)?

  1. 16t7−6t+20t316t^7-6t+20t^3 (correct answer)
  2. 16t8−6t2+20t416t^8-6t^2+20t^4
  3. 2t7−3t+5t32t^7-3t+5t^3
  4. 16t7−3t+20t316t^7-3t+20t^3
  5. 16t7−6t+5t416t^7-6t+5t^4

Explanation: This problem requires applying the power rule to a polynomial with terms out of order. The power rule states that d/dt[t^n] = nt^(n-1). For d(t) = 2t^8 - 3t^2 + 5t^4, we differentiate each term independently: the derivative of 2t^8 is 2·8t^7 = 16t^7, the derivative of -3t^2 is -3·2t^1 = -6t, and the derivative of 5t^4 is 5·4t^3 = 20t^3. Therefore, d'(t) = 16t^7 - 6t + 20t^3. Choice E incorrectly differentiates the last term as 5t^4 instead of 20t^3, keeping the original term unchanged. When differentiating polynomials, treat each term separately and don't be distracted by their order in the expression.