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  3. AP Calculus AB

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AP Calculus AB Flashcards: Applying The Power Rule

Study Applying The Power Rule in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

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What this deck covers

This deck focuses on Applying The Power Rule, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

AP Calculus AB Flashcards: Applying The Power Rule

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QUESTION

What is the derivative of f(x)=8x−4f(x) = 8x^{-4}f(x)=8x−4?

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ANSWER

f′(x)=−32x−5f'(x) = -32x^{-5}f′(x)=−32x−5. Apply Power Rule: 8⋅(−4)⋅x−4−1=−32x−58 \cdot (-4) \cdot x^{-4-1} = -32x^{-5}8⋅(−4)⋅x−4−1=−32x−5.

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Flashcard 1: What is the derivative of f(x)=8x−4f(x) = 8x^{-4}f(x)=8x−4?

Answer: f′(x)=−32x−5f'(x) = -32x^{-5}f′(x)=−32x−5. Apply Power Rule: 8⋅(−4)⋅x−4−1=−32x−58 \cdot (-4) \cdot x^{-4-1} = -32x^{-5}8⋅(−4)⋅x−4−1=−32x−5.

Flashcard 2: Differentiate f(x)=3x5/2−4x3/2f(x) = 3x^{5/2} - 4x^{3/2}f(x)=3x5/2−4x3/2 using the Power Rule.

Answer: f′(x)=152x3/2−6x1/2f'(x) = \frac{15}{2}x^{3/2} - 6x^{1/2}f′(x)=215​x3/2−6x1/2. Differentiate each term using Power Rule.

Flashcard 3: Find the derivative of f(x)=x0f(x) = x^0f(x)=x0.

Answer: f′(x)=0f'(x) = 0f′(x)=0. x0=1x^0 = 1x0=1, which is constant, so derivative is 0.

Flashcard 4: Differentiate f(x)=9x−1/2f(x) = 9x^{-1/2}f(x)=9x−1/2 using the Power Rule.

Answer: f′(x)=−4.5x−3/2f'(x) = -4.5x^{-3/2}f′(x)=−4.5x−3/2. Apply Power Rule: 9⋅(−12)⋅x−12−1=−4.5x−3/29 \cdot (-\frac{1}{2}) \cdot x^{-\frac{1}{2}-1} = -4.5x^{-3/2}9⋅(−21​)⋅x−21​−1=−4.5x−3/2.

Flashcard 5: Differentiate f(x)=x−2/3f(x) = x^{-2/3}f(x)=x−2/3 using the Power Rule.

Answer: f′(x)=−23x−5/3f'(x) = -\frac{2}{3}x^{-5/3}f′(x)=−32​x−5/3. Apply Power Rule: −23⋅x−23−1=−23x−5/3-\frac{2}{3} \cdot x^{-\frac{2}{3}-1} = -\frac{2}{3}x^{-5/3}−32​⋅x−32​−1=−32​x−5/3.

Flashcard 6: Differentiate f(x)=15x1/4f(x) = 15x^{1/4}f(x)=15x1/4 using the Power Rule.

Answer: f′(x)=154x−3/4f'(x) = \frac{15}{4}x^{-3/4}f′(x)=415​x−3/4. Apply Power Rule: 15⋅14⋅x14−1=154x−3/415 \cdot \frac{1}{4} \cdot x^{\frac{1}{4}-1} = \frac{15}{4}x^{-3/4}15⋅41​⋅x41​−1=415​x−3/4.

Flashcard 7: Differentiate f(x)=x1/3f(x) = x^{1/3}f(x)=x1/3 using the Power Rule.

Answer: f′(x)=13x−2/3f'(x) = \frac{1}{3}x^{-2/3}f′(x)=31​x−2/3. Apply Power Rule: 13⋅x13−1=13x−2/3\frac{1}{3} \cdot x^{\frac{1}{3}-1} = \frac{1}{3}x^{-2/3}31​⋅x31​−1=31​x−2/3.

Flashcard 8: Differentiate f(x)=5f(x) = 5f(x)=5 using the Power Rule.

Answer: f′(x)=0f'(x) = 0f′(x)=0. Constant functions have derivative 0.

Flashcard 9: Differentiate f(x)=x1/2f(x) = x^{1/2}f(x)=x1/2 using the Power Rule.

Answer: f′(x)=12x−1/2f'(x) = \frac{1}{2}x^{-1/2}f′(x)=21​x−1/2. Apply Power Rule: 12⋅x12−1=12x−1/2\frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{-1/2}21​⋅x21​−1=21​x−1/2.

Flashcard 10: What is the derivative of f(x)=4x0f(x) = 4x^0f(x)=4x0?

Answer: f′(x)=0f'(x) = 0f′(x)=0. 4x0=44x^0 = 44x0=4, which is constant, so derivative is 0.

Flashcard 11: Differentiate f(x)=3x4.5f(x) = 3x^{4.5}f(x)=3x4.5 using the Power Rule.

Answer: f′(x)=13.5x3.5f'(x) = 13.5x^{3.5}f′(x)=13.5x3.5. Apply Power Rule: 3⋅4.5⋅x4.5−1=13.5x3.53 \cdot 4.5 \cdot x^{4.5-1} = 13.5x^{3.5}3⋅4.5⋅x4.5−1=13.5x3.5.

Flashcard 12: Differentiate f(x)=x−3f(x) = x^{-3}f(x)=x−3 using the Power Rule.

Answer: f′(x)=−3x−4f'(x) = -3x^{-4}f′(x)=−3x−4. Apply Power Rule: −3⋅x−3−1=−3x−4-3 \cdot x^{-3-1} = -3x^{-4}−3⋅x−3−1=−3x−4.

Flashcard 13: Differentiate f(x)=x5f(x) = x^5f(x)=x5 using the Power Rule.

Answer: f′(x)=5x4f'(x) = 5x^4f′(x)=5x4. Apply Power Rule: 5⋅x5−1=5x45 \cdot x^{5-1} = 5x^45⋅x5−1=5x4.

Flashcard 14: What is the derivative of f(x)=x2/3f(x) = x^{2/3}f(x)=x2/3?

Answer: f′(x)=23x−1/3f'(x) = \frac{2}{3}x^{-1/3}f′(x)=32​x−1/3. Apply Power Rule: 23⋅x23−1=23x−1/3\frac{2}{3} \cdot x^{\frac{2}{3}-1} = \frac{2}{3}x^{-1/3}32​⋅x32​−1=32​x−1/3.

Flashcard 15: What is the derivative of f(x)=−5x0.5f(x) = -5x^{0.5}f(x)=−5x0.5?

Answer: f′(x)=−2.5x−0.5f'(x) = -2.5x^{-0.5}f′(x)=−2.5x−0.5. Apply Power Rule: −5⋅0.5⋅x0.5−1=−2.5x−0.5-5 \cdot 0.5 \cdot x^{0.5-1} = -2.5x^{-0.5}−5⋅0.5⋅x0.5−1=−2.5x−0.5.

Flashcard 16: Differentiate f(x)=8x2/5f(x) = 8x^{2/5}f(x)=8x2/5 using the Power Rule.

Answer: f′(x)=165x−3/5f'(x) = \frac{16}{5}x^{-3/5}f′(x)=516​x−3/5. Apply Power Rule: 8⋅25⋅x25−1=165x−3/58 \cdot \frac{2}{5} \cdot x^{\frac{2}{5}-1} = \frac{16}{5}x^{-3/5}8⋅52​⋅x52​−1=516​x−3/5.

Flashcard 17: Differentiate f(x)=13x−0.3f(x) = 13x^{-0.3}f(x)=13x−0.3 using the Power Rule.

Answer: f′(x)=−3.9x−1.3f'(x) = -3.9x^{-1.3}f′(x)=−3.9x−1.3. Apply Power Rule: 13⋅(−0.3)⋅x−0.3−1=−3.9x−1.313 \cdot (-0.3) \cdot x^{-0.3-1} = -3.9x^{-1.3}13⋅(−0.3)⋅x−0.3−1=−3.9x−1.3.

Flashcard 18: Find the derivative of f(x)=x3.5f(x) = x^{3.5}f(x)=x3.5 using the Power Rule.

Answer: f′(x)=3.5x2.5f'(x) = 3.5x^{2.5}f′(x)=3.5x2.5. Apply Power Rule: 3.5⋅x3.5−1=3.5x2.53.5 \cdot x^{3.5-1} = 3.5x^{2.5}3.5⋅x3.5−1=3.5x2.5.

Flashcard 19: Differentiate f(x)=x0.1f(x) = x^{0.1}f(x)=x0.1 using the Power Rule.

Answer: f′(x)=0.1x−0.9f'(x) = 0.1x^{-0.9}f′(x)=0.1x−0.9. Apply Power Rule: 0.1⋅x0.1−1=0.1x−0.90.1 \cdot x^{0.1-1} = 0.1x^{-0.9}0.1⋅x0.1−1=0.1x−0.9.

Flashcard 20: Differentiate f(x)=x6−x2f(x) = x^6 - x^2f(x)=x6−x2 using the Power Rule.

Answer: f′(x)=6x5−2xf'(x) = 6x^5 - 2xf′(x)=6x5−2x. Differentiate each term: 6x5−2x6x^5 - 2x6x5−2x.

Flashcard 21: What is the derivative of f(x)=x9f(x) = x^9f(x)=x9?

Answer: f′(x)=9x8f'(x) = 9x^8f′(x)=9x8. Apply Power Rule: 9⋅x9−1=9x89 \cdot x^{9-1} = 9x^89⋅x9−1=9x8.

Flashcard 22: Find the derivative of f(x)=7x3/7f(x) = 7x^{3/7}f(x)=7x3/7 using the Power Rule.

Answer: f′(x)=3x−4/7f'(x) = 3x^{-4/7}f′(x)=3x−4/7. Apply Power Rule: 7⋅37⋅x37−1=3x−4/77 \cdot \frac{3}{7} \cdot x^{\frac{3}{7}-1} = 3x^{-4/7}7⋅73​⋅x73​−1=3x−4/7.

Flashcard 23: What is the derivative of f(x)=−2x3f(x) = -2x^3f(x)=−2x3?

Answer: f′(x)=−6x2f'(x) = -6x^2f′(x)=−6x2. Apply Power Rule: −2⋅3⋅x3−1=−6x2-2 \cdot 3 \cdot x^{3-1} = -6x^2−2⋅3⋅x3−1=−6x2.

Flashcard 24: Find f′(x)f'(x)f′(x) for f(x)=10x−0.5f(x) = 10x^{-0.5}f(x)=10x−0.5. Use the Power Rule.

Answer: f′(x)=−5x−1.5f'(x) = -5x^{-1.5}f′(x)=−5x−1.5. Apply Power Rule: 10⋅(−0.5)⋅x−0.5−1=−5x−1.510 \cdot (-0.5) \cdot x^{-0.5-1} = -5x^{-1.5}10⋅(−0.5)⋅x−0.5−1=−5x−1.5.

Flashcard 25: Derive f(x)=x−1f(x) = x^{-1}f(x)=x−1 using the Power Rule.

Answer: f′(x)=−x−2f'(x) = -x^{-2}f′(x)=−x−2. Apply Power Rule: −1⋅x−1−1=−x−2-1 \cdot x^{-1-1} = -x^{-2}−1⋅x−1−1=−x−2.

Flashcard 26: Differentiate f(x)=6x3+7xf(x) = 6x^3 + 7xf(x)=6x3+7x using the Power Rule.

Answer: f′(x)=18x2+7f'(x) = 18x^2 + 7f′(x)=18x2+7. Differentiate each term: 6⋅3x2+7⋅1=18x2+76 \cdot 3x^2 + 7 \cdot 1 = 18x^2 + 76⋅3x2+7⋅1=18x2+7.

Flashcard 27: Find the derivative of f(x)=6x−3/2f(x) = 6x^{-3/2}f(x)=6x−3/2 using the Power Rule.

Answer: f′(x)=−9x−5/2f'(x) = -9x^{-5/2}f′(x)=−9x−5/2. Apply Power Rule: 6⋅(−32)⋅x−32−1=−9x−5/26 \cdot (-\frac{3}{2}) \cdot x^{-\frac{3}{2}-1} = -9x^{-5/2}6⋅(−23​)⋅x−23​−1=−9x−5/2.

Flashcard 28: Differentiate f(x)=x10f(x) = x^{10}f(x)=x10 using the Power Rule.

Answer: f′(x)=10x9f'(x) = 10x^9f′(x)=10x9. Apply Power Rule: 10⋅x10−1=10x910 \cdot x^{10-1} = 10x^910⋅x10−1=10x9.

Flashcard 29: Differentiate f(x)=11x−2/5f(x) = 11x^{-2/5}f(x)=11x−2/5 using the Power Rule.

Answer: f′(x)=−225x−7/5f'(x) = -\frac{22}{5}x^{-7/5}f′(x)=−522​x−7/5. Apply Power Rule: 11⋅(−25)⋅x−25−1=−225x−7/511 \cdot (-\frac{2}{5}) \cdot x^{-\frac{2}{5}-1} = -\frac{22}{5}x^{-7/5}11⋅(−52​)⋅x−52​−1=−522​x−7/5.

Flashcard 30: Find the derivative of f(x)=x3.3f(x) = x^{3.3}f(x)=x3.3 using the Power Rule.

Answer: f′(x)=3.3x2.3f'(x) = 3.3x^{2.3}f′(x)=3.3x2.3. Apply Power Rule: 3.3⋅x3.3−1=3.3x2.33.3 \cdot x^{3.3-1} = 3.3x^{2.3}3.3⋅x3.3−1=3.3x2.3.