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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.

AP Calculus BC

Applying The Power Rule

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

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ANSWER

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

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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.

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Flashcard 1: Differentiate: f(x)=5x3.5f(x) = 5x^{3.5}.

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

Flashcard 2: Differentiate: f(x)=8x3f(x) = 8x^{3}.

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

Flashcard 3: Find the derivative of x12x^{\frac{1}{2}}.

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

Flashcard 4: What is the derivative of x2.7x^{2.7}?

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

Flashcard 5: Differentiate: f(x)=6x2f(x) = 6x^2.

Answer: f′(x)=12xf'(x) = 12x. Factor out 6, apply power rule: 6⋅2x16 \cdot 2x^1.

Flashcard 6: Differentiate: f(x)=7x0f(x) = 7x^{0}.

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

Flashcard 7: Differentiate: f(x)=6x2f(x) = 6x^2.

Answer: f′(x)=12xf'(x) = 12x. Factor out 6, apply power rule: 6⋅2x16 \cdot 2x^1.

Flashcard 8: What is the derivative of x9x^9?

Answer: 9x89x^8. Power rule: 9⋅x9−1=9x89 \cdot x^{9-1} = 9x^8.

Flashcard 9: What is the derivative of x94x^{\frac{9}{4}}?

Answer: 94x54\frac{9}{4}x^{\frac{5}{4}}. Power rule: 94⋅x94−1=94x54\frac{9}{4} \cdot x^{\frac{9}{4}-1} = \frac{9}{4}x^{\frac{5}{4}}.

Flashcard 10: Differentiate: f(x)=−3x4.5f(x) = -3x^{4.5}.

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

Flashcard 11: What is the derivative of x13x^{\frac{1}{3}}?

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

Flashcard 12: Differentiate: f(x)=2x−14f(x) = 2x^{-\frac{1}{4}}.

Answer: f′(x)=−12x−54f'(x) = -\frac{1}{2}x^{-\frac{5}{4}}. Factor out 2, apply power rule: 2⋅(−14)x−542 \cdot (-\frac{1}{4})x^{-\frac{5}{4}}.

Flashcard 13: What is the derivative of x0x^0?

Answer: 00. Since x0=1x^0 = 1 (a constant), its derivative is 0.

Flashcard 14: What is the derivative of a constant cc?

Answer: 00. The derivative of any constant is always zero.

Flashcard 15: What is the derivative of x5x^5?

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

Flashcard 16: Differentiate: f(x)=x3f(x) = x^3.

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

Flashcard 17: What is the derivative of x0.1x^{0.1}?

Answer: 0.1x−0.90.1x^{-0.9}. Power rule: 0.1⋅x0.1−1=0.1x−0.90.1 \cdot x^{0.1-1} = 0.1x^{-0.9}.

Flashcard 18: What is the derivative of x10x^{10}?

Answer: 10x910x^9. Power rule: 10⋅x10−1=10x910 \cdot x^{10-1} = 10x^9.

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

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

Flashcard 20: What is the derivative of x−5x^{-5}?

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

Flashcard 21: What is the derivative of x−7x^{-7}?

Answer: −7x−8-7x^{-8}. Power rule: −7⋅x−7−1=−7x−8-7 \cdot x^{-7-1} = -7x^{-8}.

Flashcard 22: What is the derivative of x1.5x^{1.5}?

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

Flashcard 23: Differentiate: f(x)=x45f(x) = x^{\frac{4}{5}}.

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

Flashcard 24: State the derivative of x32x^{\frac{3}{2}}.

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

Flashcard 25: What is the derivative of x72x^{\frac{7}{2}}?

Answer: 72x52\frac{7}{2}x^{\frac{5}{2}}. Power rule: 72⋅x72−1=72x52\frac{7}{2} \cdot x^{\frac{7}{2}-1} = \frac{7}{2}x^{\frac{5}{2}}.

Flashcard 26: Differentiate: f(x)=−5x6f(x) = -5x^{6}.

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

Flashcard 27: What is the derivative of x5x^5?

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

Flashcard 28: What is the derivative of x10x^{10}?

Answer: 10x910x^9. Power rule: 10⋅x10−1=10x910 \cdot x^{10-1} = 10x^9.

Flashcard 29: What is the derivative of x13x^{\frac{1}{3}}?

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

Flashcard 30: State the Power Rule for differentiation of xnx^n.

Answer: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}. The fundamental rule for differentiating power functions.

Flashcard 31: What is the derivative of x2.7x^{2.7}?

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

Flashcard 32: Differentiate: f(x)=x−12f(x) = x^{-\frac{1}{2}}.

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

Flashcard 33: Differentiate: f(x)=8x3f(x) = 8x^{3}.

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

Flashcard 34: What is the derivative of x43x^{\frac{4}{3}}?

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

Flashcard 35: Differentiate: f(x)=−5x6f(x) = -5x^{6}.

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

Flashcard 36: Differentiate: f(x)=7x4f(x) = 7x^4.

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

Flashcard 37: Differentiate: f(x)=x3f(x) = x^3.

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

Flashcard 38: What is the derivative of x53x^{\frac{5}{3}}?

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

Flashcard 39: Differentiate: f(x)=7x0f(x) = 7x^{0}.

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

Flashcard 40: Differentiate: f(x)=−x8f(x) = -x^8.

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

Flashcard 41: Differentiate: f(x)=3x23f(x) = 3x^{\frac{2}{3}}.

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

Flashcard 42: Differentiate: f(x)=7x4f(x) = 7x^4.

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

Flashcard 43: State the Power Rule for differentiation of xnx^n.

Answer: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}. The fundamental rule for differentiating power functions.

Flashcard 44: What is the derivative of x−1x^{-1}?

Answer: −x−2-x^{-2}. Power rule: −1⋅x−1−1=−x−2-1 \cdot x^{-1-1} = -x^{-2}.

Flashcard 45: Differentiate: f(x)=−4x6f(x) = -4x^6.

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

Flashcard 46: Differentiate: f(x)=x−2f(x) = x^{-2}.

Answer: f′(x)=−2x−3f'(x) = -2x^{-3}. Power rule with negative exponent: −2⋅x−2−1-2 \cdot x^{-2-1}.

Flashcard 47: Differentiate: f(x)=10x0.5f(x) = 10x^{0.5}.

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

Flashcard 48: Differentiate: f(x)=9x−3f(x) = 9x^{-3}.

Answer: f′(x)=−27x−4f'(x) = -27x^{-4}. Factor out constant 9, then apply power rule: 9⋅(−3)x−49 \cdot (-3)x^{-4}.

Flashcard 49: What is the derivative of x53x^{\frac{5}{3}}?

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

Flashcard 50: What is the derivative of x72x^{\frac{7}{2}}?

Answer: 72x52\frac{7}{2}x^{\frac{5}{2}}. Power rule: 72⋅x72−1=72x52\frac{7}{2} \cdot x^{\frac{7}{2}-1} = \frac{7}{2}x^{\frac{5}{2}}.

Flashcard 51: Differentiate: f(x)=10x0.5f(x) = 10x^{0.5}.

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

Flashcard 52: What is the derivative of x−7x^{-7}?

Answer: −7x−8-7x^{-8}. Power rule: −7⋅x−7−1=−7x−8-7 \cdot x^{-7-1} = -7x^{-8}.

Flashcard 53: Differentiate: f(x)=9x−3f(x) = 9x^{-3}.

Answer: f′(x)=−27x−4f'(x) = -27x^{-4}. Factor out constant 9, then apply power rule: 9⋅(−3)x−49 \cdot (-3)x^{-4}.

Flashcard 54: Differentiate: f(x)=2x−14f(x) = 2x^{-\frac{1}{4}}.

Answer: f′(x)=−12x−54f'(x) = -\frac{1}{2}x^{-\frac{5}{4}}. Factor out 2, apply power rule: 2⋅(−14)x−542 \cdot (-\frac{1}{4})x^{-\frac{5}{4}}.

Flashcard 55: What is the derivative of x1x^1?

Answer: 11. The derivative of xx (first power) is 1.

Flashcard 56: Differentiate: f(x)=x−10f(x) = x^{-10}.

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

Flashcard 57: Differentiate: f(x)=4x2.3f(x) = 4x^{2.3}.

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

Flashcard 58: Differentiate: f(x)=−4x6f(x) = -4x^6.

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

Flashcard 59: What is the derivative of x1.5x^{1.5}?

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

Flashcard 60: Differentiate: f(x)=4x2.3f(x) = 4x^{2.3}.

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

Flashcard 61: What is the derivative of x0x^0?

Answer: 00. Since x0=1x^0 = 1 (a constant), its derivative is 0.

Flashcard 62: Differentiate: f(x)=5x3.5f(x) = 5x^{3.5}.

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

Flashcard 63: What is the derivative of x94x^{\frac{9}{4}}?

Answer: 94x54\frac{9}{4}x^{\frac{5}{4}}. Power rule: 94⋅x94−1=94x54\frac{9}{4} \cdot x^{\frac{9}{4}-1} = \frac{9}{4}x^{\frac{5}{4}}.

Flashcard 64: Differentiate: f(x)=x−12f(x) = x^{-\frac{1}{2}}.

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

Flashcard 65: What is the derivative of x0.1x^{0.1}?

Answer: 0.1x−0.90.1x^{-0.9}. Power rule: 0.1⋅x0.1−1=0.1x−0.90.1 \cdot x^{0.1-1} = 0.1x^{-0.9}.

Flashcard 66: What is the derivative of x1x^1?

Answer: 11. The derivative of xx (first power) is 1.

Flashcard 67: State the derivative of x32x^{\frac{3}{2}}.

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

Flashcard 68: What is the derivative of x43x^{\frac{4}{3}}?

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

Flashcard 69: What is the derivative of x−1x^{-1}?

Answer: −x−2-x^{-2}. Power rule: −1⋅x−1−1=−x−2-1 \cdot x^{-1-1} = -x^{-2}.

Flashcard 70: What is the derivative of x−5x^{-5}?

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

Flashcard 71: Differentiate: f(x)=−x8f(x) = -x^8.

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

Flashcard 72: What is the derivative of x9x^9?

Answer: 9x89x^8. Power rule: 9⋅x9−1=9x89 \cdot x^{9-1} = 9x^8.

Flashcard 73: What is the derivative of a constant cc?

Answer: 00. The derivative of any constant is always zero.

Flashcard 74: Differentiate: f(x)=x−2f(x) = x^{-2}.

Answer: f′(x)=−2x−3f'(x) = -2x^{-3}. Power rule with negative exponent: −2⋅x−2−1-2 \cdot x^{-2-1}.

Flashcard 75: Differentiate: f(x)=3x23f(x) = 3x^{\frac{2}{3}}.

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

Flashcard 76: Differentiate: f(x)=−3x4.5f(x) = -3x^{4.5}.

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

Flashcard 77: Find the derivative of x12x^{\frac{1}{2}}.

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

Flashcard 78: Differentiate: f(x)=x45f(x) = x^{\frac{4}{5}}.

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