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AP Calculus AB Flashcards: Calculating Higher Order Derivatives

Study Calculating Higher Order Derivatives 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 Calculating Higher Order Derivatives, 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: Calculating Higher Order Derivatives

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QUESTION

Find the second derivative of f(x)=e2xf(x) = e^{2x}f(x)=e2x.

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ANSWER

f′′(x)=4e2xf''(x) = 4e^{2x}f′′(x)=4e2x. Use chain rule: f′(x)=2e2xf'(x) = 2e^{2x}f′(x)=2e2x, then f′′(x)=4e2xf''(x) = 4e^{2x}f′′(x)=4e2x.

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Flashcard 1: Find the second derivative of f(x)=e2xf(x) = e^{2x}f(x)=e2x.

Answer: f′′(x)=4e2xf''(x) = 4e^{2x}f′′(x)=4e2x. Use chain rule: f′(x)=2e2xf'(x) = 2e^{2x}f′(x)=2e2x, then f′′(x)=4e2xf''(x) = 4e^{2x}f′′(x)=4e2x.

Flashcard 2: Find the third derivative of f(x)=5x4+3x2f(x) = 5x^4 + 3x^2f(x)=5x4+3x2.

Answer: f(3)(x)=120xf^{(3)}(x) = 120xf(3)(x)=120x. Differentiate three times: f′(x)=20x3+6xf'(x) = 20x^3 + 6xf′(x)=20x3+6x, f′′(x)=60x2+6f''(x) = 60x^2 + 6f′′(x)=60x2+6, f′′′(x)=120xf'''(x) = 120xf′′′(x)=120x.

Flashcard 3: State the formula for the nnn-th derivative of f(x)=xnf(x) = x^nf(x)=xn.

Answer: f(n)(x)=n!f^{(n)}(x) = n!f(n)(x)=n! if nnn is a positive integer. Each differentiation reduces the power by 1 and multiplies by the current power.

Flashcard 4: What is the second derivative of f(x)=exf(x) = e^xf(x)=ex?

Answer: f′′(x)=exf''(x) = e^xf′′(x)=ex. The exponential function exe^xex is its own derivative.

Flashcard 5: Calculate the second derivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​.

Answer: f′′(x)=2x3f''(x) = \frac{2}{x^3}f′′(x)=x32​. Rewrite as x−1x^{-1}x−1, apply power rule: f′(x)=−x−2f'(x) = -x^{-2}f′(x)=−x−2, f′′(x)=2x−3f''(x) = 2x^{-3}f′′(x)=2x−3.

Flashcard 6: Identify the second derivative of f(x)=12x2−x+1f(x) = \frac{1}{2}x^2 - x + 1f(x)=21​x2−x+1.

Answer: f′′(x)=1f''(x) = 1f′′(x)=1. First derivative is x−1x - 1x−1, second derivative is constant 111.

Flashcard 7: Calculate the second derivative of f(x)=3x3−x2+xf(x) = 3x^3 - x^2 + xf(x)=3x3−x2+x.

Answer: f′′(x)=18x−2f''(x) = 18x - 2f′′(x)=18x−2. First derivative is 9x2−2x+19x^2 - 2x + 19x2−2x+1, second derivative is 18x−218x - 218x−2.

Flashcard 8: What is the second derivative of f(x)=15x5−13x3f(x) = \frac{1}{5}x^5 - \frac{1}{3}x^3f(x)=51​x5−31​x3?

Answer: f′′(x)=4x3−2xf''(x) = 4x^3 - 2xf′′(x)=4x3−2x. Apply power rule twice to each term separately.

Flashcard 9: Calculate the second derivative of f(x)=14x4+xf(x) = \frac{1}{4}x^4 + xf(x)=41​x4+x.

Answer: f′′(x)=3x2f''(x) = 3x^2f′′(x)=3x2. First derivative is x3+1x^3 + 1x3+1, second derivative is 3x23x^23x2.

Flashcard 10: What is the second derivative of f(x)=12x2f(x) = \frac{1}{2}x^2f(x)=21​x2?

Answer: f′′(x)=1f''(x) = 1f′′(x)=1. First derivative is xxx, second derivative is constant 111.

Flashcard 11: Calculate the second derivative of f(x)=13x3+2xf(x) = \frac{1}{3}x^3 + 2xf(x)=31​x3+2x.

Answer: f′′(x)=2xf''(x) = 2xf′′(x)=2x. First derivative is x2+2x^2 + 2x2+2, second derivative is 2x2x2x.

Flashcard 12: Calculate the second derivative of f(x)=4x4−2x2f(x) = 4x^4 - 2x^2f(x)=4x4−2x2.

Answer: f′′(x)=48x2−4f''(x) = 48x^2 - 4f′′(x)=48x2−4. Apply power rule twice: f′(x)=16x3−4xf'(x) = 16x^3 - 4xf′(x)=16x3−4x, then f′′(x)=48x2−4f''(x) = 48x^2 - 4f′′(x)=48x2−4.

Flashcard 13: What is the second derivative of f(x)=x2−4x+4f(x) = x^2 - 4x + 4f(x)=x2−4x+4?

Answer: f′′(x)=2f''(x) = 2f′′(x)=2. First derivative is 2x−42x - 42x−4, second derivative is constant 222.

Flashcard 14: Find the second derivative of f(x)=x3−3x2+3x−1f(x) = x^3 - 3x^2 + 3x - 1f(x)=x3−3x2+3x−1.

Answer: f′′(x)=6x−6f''(x) = 6x - 6f′′(x)=6x−6. First derivative is 3x2−6x+33x^2 - 6x + 33x2−6x+3, second derivative is 6x−66x - 66x−6.

Flashcard 15: What is the second derivative of f(x)=x4−2x2+1f(x) = x^4 - 2x^2 + 1f(x)=x4−2x2+1?

Answer: f′′(x)=12x2−4f''(x) = 12x^2 - 4f′′(x)=12x2−4. First derivative is 4x3−4x4x^3 - 4x4x3−4x, second derivative is 12x2−412x^2 - 412x2−4.

Flashcard 16: Find the second derivative of f(x)=e3xf(x) = e^{3x}f(x)=e3x.

Answer: f′′(x)=9e3xf''(x) = 9e^{3x}f′′(x)=9e3x. Use chain rule: f′(x)=3e3xf'(x) = 3e^{3x}f′(x)=3e3x, then f′′(x)=9e3xf''(x) = 9e^{3x}f′′(x)=9e3x.

Flashcard 17: Calculate the third derivative of f(x)=x5−x3+xf(x) = x^5 - x^3 + xf(x)=x5−x3+x.

Answer: f(3)(x)=60x2−6f^{(3)}(x) = 60x^2 - 6f(3)(x)=60x2−6. Differentiate three times: f′(x)=5x4−3x2+1f'(x) = 5x^4 - 3x^2 + 1f′(x)=5x4−3x2+1, then continue.

Flashcard 18: State the second derivative of f(x)=14x4−x2f(x) = \frac{1}{4}x^4 - x^2f(x)=41​x4−x2.

Answer: f′′(x)=3x2−2f''(x) = 3x^2 - 2f′′(x)=3x2−2. Apply power rule twice: f′(x)=x3−2xf'(x) = x^3 - 2xf′(x)=x3−2x, then f′′(x)=3x2−2f''(x) = 3x^2 - 2f′′(x)=3x2−2.

Flashcard 19: What is the second derivative of f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1?

Answer: f′′(x)=2f''(x) = 2f′′(x)=2. First derivative is 2x+22x + 22x+2, second derivative is constant 222.

Flashcard 20: What is the second derivative of f(x)=x3+3x2+3x+1f(x) = x^3 + 3x^2 + 3x + 1f(x)=x3+3x2+3x+1?

Answer: f′′(x)=6x+6f''(x) = 6x + 6f′′(x)=6x+6. First derivative is 3x2+6x+33x^2 + 6x + 33x2+6x+3, second derivative is 6x+66x + 66x+6.

Flashcard 21: Calculate the third derivative of f(x)=16x6f(x) = \frac{1}{6}x^6f(x)=61​x6.

Answer: f(3)(x)=20x3f^{(3)}(x) = 20x^3f(3)(x)=20x3. Apply power rule three times: f′(x)=x5f'(x) = x^5f′(x)=x5, f′′(x)=5x4f''(x) = 5x^4f′′(x)=5x4, f′′′(x)=20x3f'''(x) = 20x^3f′′′(x)=20x3.

Flashcard 22: Find the second derivative of f(x)=x5+x4+x3f(x) = x^5 + x^4 + x^3f(x)=x5+x4+x3.

Answer: f′′(x)=20x3+12x2+6xf''(x) = 20x^3 + 12x^2 + 6xf′′(x)=20x3+12x2+6x. Apply power rule twice to each term separately.

Flashcard 23: State the second derivative of f(x)=12x2−xf(x) = \frac{1}{2}x^2 - xf(x)=21​x2−x.

Answer: f′′(x)=1f''(x) = 1f′′(x)=1. First derivative is x−1x - 1x−1, second derivative is constant 111.

Flashcard 24: What is the third derivative of f(x)=13x3+x2f(x) = \frac{1}{3}x^3+x^2f(x)=31​x3+x2?

Answer: f(3)(x)=2f^{(3)}(x) = 2f(3)(x)=2. First derivative is x2+2xx^2 + 2xx2+2x, second is 2x+22x + 22x+2, third is 222.

Flashcard 25: Calculate the second derivative of f(x)=3x3−3x2+3x−1f(x) = 3x^3 - 3x^2 + 3x - 1f(x)=3x3−3x2+3x−1.

Answer: f′′(x)=18x−6f''(x) = 18x - 6f′′(x)=18x−6. First derivative is 9x2−6x+39x^2 - 6x + 39x2−6x+3, second derivative is 18x−618x - 618x−6.

Flashcard 26: Find the second derivative of f(x)=13x3f(x) = \frac{1}{3}x^3f(x)=31​x3.

Answer: f′′(x)=2xf''(x) = 2xf′′(x)=2x. First derivative is x2x^2x2, second derivative is 2x2x2x.

Flashcard 27: What is the third derivative of f(x)=7x5−3x4+x3f(x) = 7x^5 - 3x^4 + x^3f(x)=7x5−3x4+x3?

Answer: f(3)(x)=420x2−72x+6f^{(3)}(x) = 420x^2 - 72x + 6f(3)(x)=420x2−72x+6. Differentiate each term three times using power rule.

Flashcard 28: What is the second derivative of f(x)=15x5f(x) = \frac{1}{5}x^5f(x)=51​x5?

Answer: f′′(x)=4x3f''(x) = 4x^3f′′(x)=4x3. Apply power rule twice: f′(x)=x4f'(x) = x^4f′(x)=x4, then f′′(x)=4x3f''(x) = 4x^3f′′(x)=4x3.

Flashcard 29: State the second derivative of f(x)=13x3−x2+1f(x) = \frac{1}{3}x^3 - x^2 + 1f(x)=31​x3−x2+1.

Answer: f′′(x)=2x−2f''(x) = 2x - 2f′′(x)=2x−2. First derivative is x2−2xx^2 - 2xx2−2x, second derivative is 2x−22x - 22x−2.

Flashcard 30: State the second derivative of f(x)=12x2+3x+5f(x) = \frac{1}{2}x^2 + 3x + 5f(x)=21​x2+3x+5.

Answer: f′′(x)=1f''(x) = 1f′′(x)=1. First derivative is x+3x + 3x+3, second derivative is constant 111.