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AP Calculus BC Flashcards: Applying The Power Rule

Study Applying The Power Rule in AP Calculus BC 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 BC.

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 BC Flashcards: Applying The Power Rule

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QUESTION

What is the derivative of x1x^1x1?

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ANSWER

111. The derivative of xxx (first power) is 1.

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All flashcards

Flashcard 1: What is the derivative of x1x^1x1?

Answer: 111. The derivative of xxx (first power) is 1.

Flashcard 2: What is the derivative of x10x^{10}x10?

Answer: 10x910x^910x9. Power rule: 10⋅x10−1=10x910 \cdot x^{10-1} = 10x^910⋅x10−1=10x9.

Flashcard 3: Differentiate: f(x)=4x2.3f(x) = 4x^{2.3}f(x)=4x2.3.

Answer: f′(x)=9.2x1.3f'(x) = 9.2x^{1.3}f′(x)=9.2x1.3. Factor out 4, apply power rule: 4⋅2.3x1.34 \cdot 2.3x^{1.3}4⋅2.3x1.3.

Flashcard 4: What is the derivative of x53x^{\frac{5}{3}}x35​?

Answer: 53x23\frac{5}{3}x^{\frac{2}{3}}35​x32​. Power rule: 53⋅x53−1=53x23\frac{5}{3} \cdot x^{\frac{5}{3}-1} = \frac{5}{3}x^{\frac{2}{3}}35​⋅x35​−1=35​x32​.

Flashcard 5: Find the derivative of x12x^{\frac{1}{2}}x21​.

Answer: 12x−12\frac{1}{2}x^{-\frac{1}{2}}21​x−21​. Power rule: 12⋅x12−1=12x−12\frac{1}{2} \cdot x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}}21​⋅x21​−1=21​x−21​.

Flashcard 6: Differentiate: f(x)=3x23f(x) = 3x^{\frac{2}{3}}f(x)=3x32​.

Answer: f′(x)=2x−13f'(x) = 2x^{-\frac{1}{3}}f′(x)=2x−31​. Factor out 3, apply power rule: 3⋅23x23−13 \cdot \frac{2}{3}x^{\frac{2}{3}-1}3⋅32​x32​−1.

Flashcard 7: What is the derivative of x5x^5x5?

Answer: 5x45x^45x4. Power rule: n⋅xn−1=5⋅x5−1=5x4n \cdot x^{n-1} = 5 \cdot x^{5-1} = 5x^4n⋅xn−1=5⋅x5−1=5x4.

Flashcard 8: Differentiate: f(x)=6x2f(x) = 6x^2f(x)=6x2.

Answer: f′(x)=12xf'(x) = 12xf′(x)=12x. Factor out 6, apply power rule: 6⋅2x16 \cdot 2x^16⋅2x1.

Flashcard 9: Differentiate: f(x)=7x4f(x) = 7x^4f(x)=7x4.

Answer: f′(x)=28x3f'(x) = 28x^3f′(x)=28x3. Factor out constant 7, then apply power rule: 7⋅4x37 \cdot 4x^37⋅4x3.

Flashcard 10: Differentiate: f(x)=x3f(x) = x^3f(x)=x3.

Answer: f′(x)=3x2f'(x) = 3x^2f′(x)=3x2. Apply power rule: bring down exponent 3, subtract 1 from exponent.

Flashcard 11: Differentiate: f(x)=−4x6f(x) = -4x^6f(x)=−4x6.

Answer: f′(x)=−24x5f'(x) = -24x^5f′(x)=−24x5. Factor out −4-4−4, apply power rule: −4⋅6x5-4 \cdot 6x^5−4⋅6x5.

Flashcard 12: What is the derivative of a constant ccc?

Answer: 000. The derivative of any constant is always zero.

Flashcard 13: Differentiate: f(x)=x45f(x) = x^{\frac{4}{5}}f(x)=x54​.

Answer: f′(x)=45x−15f'(x) = \frac{4}{5}x^{-\frac{1}{5}}f′(x)=54​x−51​. Power rule: 45⋅x45−1=45x−15\frac{4}{5} \cdot x^{\frac{4}{5}-1} = \frac{4}{5}x^{-\frac{1}{5}}54​⋅x54​−1=54​x−51​.

Flashcard 14: What is the derivative of x2.7x^{2.7}x2.7?

Answer: 2.7x1.72.7x^{1.7}2.7x1.7. Power rule: 2.7⋅x2.7−1=2.7x1.72.7 \cdot x^{2.7-1} = 2.7x^{1.7}2.7⋅x2.7−1=2.7x1.7.

Flashcard 15: Differentiate: f(x)=7x0f(x) = 7x^{0}f(x)=7x0.

Answer: f′(x)=0f'(x) = 0f′(x)=0. Since 7x0=77x^0 = 77x0=7 (a constant), its derivative is 0.

Flashcard 16: What is the derivative of x13x^{\frac{1}{3}}x31​?

Answer: 13x−23\frac{1}{3}x^{-\frac{2}{3}}31​x−32​. Power rule: 13⋅x13−1=13x−23\frac{1}{3} \cdot x^{\frac{1}{3}-1} = \frac{1}{3}x^{-\frac{2}{3}}31​⋅x31​−1=31​x−32​.

Flashcard 17: Differentiate: f(x)=x−12f(x) = x^{-\frac{1}{2}}f(x)=x−21​.

Answer: f′(x)=−12x−32f'(x) = -\frac{1}{2}x^{-\frac{3}{2}}f′(x)=−21​x−23​. Power rule: −12⋅x−12−1=−12x−32-\frac{1}{2} \cdot x^{-\frac{1}{2}-1} = -\frac{1}{2}x^{-\frac{3}{2}}−21​⋅x−21​−1=−21​x−23​.

Flashcard 18: Differentiate: f(x)=8x3f(x) = 8x^{3}f(x)=8x3.

Answer: f′(x)=24x2f'(x) = 24x^{2}f′(x)=24x2. Factor out 8, apply power rule: 8⋅3x28 \cdot 3x^28⋅3x2.

Flashcard 19: Differentiate: f(x)=x−10f(x) = x^{-10}f(x)=x−10.

Answer: f′(x)=−10x−11f'(x) = -10x^{-11}f′(x)=−10x−11. Power rule: −10⋅x−10−1=−10x−11-10 \cdot x^{-10-1} = -10x^{-11}−10⋅x−10−1=−10x−11.

Flashcard 20: Differentiate: f(x)=−3x4.5f(x) = -3x^{4.5}f(x)=−3x4.5.

Answer: f′(x)=−13.5x3.5f'(x) = -13.5x^{3.5}f′(x)=−13.5x3.5. Factor out −3-3−3, apply power rule: −3⋅4.5x3.5-3 \cdot 4.5x^{3.5}−3⋅4.5x3.5.

Flashcard 21: What is the derivative of x0.1x^{0.1}x0.1?

Answer: 0.1x−0.90.1x^{-0.9}0.1x−0.9. 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 22: What is the derivative of x9x^9x9?

Answer: 9x89x^89x8. Power rule: 9⋅x9−1=9x89 \cdot x^{9-1} = 9x^89⋅x9−1=9x8.

Flashcard 23: Differentiate: f(x)=−x8f(x) = -x^8f(x)=−x8.

Answer: f′(x)=−8x7f'(x) = -8x^7f′(x)=−8x7. Factor out −1-1−1, apply power rule: −1⋅8x7-1 \cdot 8x^7−1⋅8x7.

Flashcard 24: Differentiate: f(x)=10x0.5f(x) = 10x^{0.5}f(x)=10x0.5.

Answer: f′(x)=5x−0.5f'(x) = 5x^{-0.5}f′(x)=5x−0.5. Factor out 10, apply power rule: 10⋅0.5x−0.510 \cdot 0.5x^{-0.5}10⋅0.5x−0.5.

Flashcard 25: Differentiate: f(x)=5x3.5f(x) = 5x^{3.5}f(x)=5x3.5.

Answer: f′(x)=17.5x2.5f'(x) = 17.5x^{2.5}f′(x)=17.5x2.5. Factor out 5, apply power rule: 5⋅3.5x2.55 \cdot 3.5x^{2.5}5⋅3.5x2.5.

Flashcard 26: State the derivative of x32x^{\frac{3}{2}}x23​.

Answer: 32x12\frac{3}{2}x^{\frac{1}{2}}23​x21​. Power rule: 32⋅x32−1=32x12\frac{3}{2} \cdot x^{\frac{3}{2}-1} = \frac{3}{2}x^{\frac{1}{2}}23​⋅x23​−1=23​x21​.

Flashcard 27: What is the derivative of x−5x^{-5}x−5?

Answer: −5x−6-5x^{-6}−5x−6. Power rule: −5⋅x−5−1=−5x−6-5 \cdot x^{-5-1} = -5x^{-6}−5⋅x−5−1=−5x−6.

Flashcard 28: Differentiate: f(x)=−5x6f(x) = -5x^{6}f(x)=−5x6.

Answer: f′(x)=−30x5f'(x) = -30x^{5}f′(x)=−30x5. Factor out −5-5−5, apply power rule: −5⋅6x5-5 \cdot 6x^5−5⋅6x5.

Flashcard 29: What is the derivative of x1.5x^{1.5}x1.5?

Answer: 1.5x0.51.5x^{0.5}1.5x0.5. Power rule: 1.5⋅x1.5−1=1.5x0.51.5 \cdot x^{1.5-1} = 1.5x^{0.5}1.5⋅x1.5−1=1.5x0.5.

Flashcard 30: What is the derivative of x43x^{\frac{4}{3}}x34​?

Answer: 43x13\frac{4}{3}x^{\frac{1}{3}}34​x31​. Power rule: 43⋅x43−1=43x13\frac{4}{3} \cdot x^{\frac{4}{3}-1} = \frac{4}{3}x^{\frac{1}{3}}34​⋅x34​−1=34​x31​.