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.

AP Calculus AB

Calculating Higher Order Derivatives

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QUESTION

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What is the second derivative of $f(x) = x^3$ ?

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ANSWER

$f''(x) = 6x$ . Apply power rule twice: $f'(x) = 3x^2$ , then $f''(x) = 6x$ .

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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.

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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: What is the second derivative of $f(x) = x^3$ ?

Answer: $f''(x) = 6x$ . Apply power rule twice: $f'(x) = 3x^2$ , then $f''(x) = 6x$ .

Flashcard 2: Calculate the third derivative of $f(x) = x^5 - x^3 + x$ .

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

Flashcard 3: Calculate the second derivative of $f(x) = \frac{1}{4}x^4 + x$ .

Answer: $f''(x) = 3x^2$ . First derivative is $x^3 + 1$ , second derivative is $3x^2$ .

Flashcard 4: State the second derivative of $f(x) = \frac{1}{2}x^2 - x$ .

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

Flashcard 5: What is the second derivative of $f(x) = x^2 + 2x + 1$ ?

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

Flashcard 6: Find the second derivative of $f(x) = e^{2x}$ .

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

Flashcard 7: State the second derivative of $f(x) = \frac{1}{2}x^2 - x$ .

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

Flashcard 8: Find the second derivative of $f(x) = x^3 - 3x^2 + 3x - 1$ .

Answer: $f''(x) = 6x - 6$ . First derivative is $3x^2 - 6x + 3$ , second derivative is $6x - 6$ .

Flashcard 9: What is the third derivative of $f(x) = \frac{1}{3}x^3+x^2$ ?

Answer: $f^{(3)}(x) = 2$ . First derivative is $x^2 + 2x$ , second is $2x + 2$ , third is $2$ .

Flashcard 10: Calculate the third derivative of $f(x) = x^5 - x^3 + x$ .

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

Flashcard 11: What is the second derivative of $f(x) = x^4 - 2x^2 + 1$ ?

Answer: $f''(x) = 12x^2 - 4$ . First derivative is $4x^3 - 4x$ , second derivative is $12x^2 - 4$ .

Flashcard 12: Calculate the second derivative of $f(x) = \frac{1}{x}$ .

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

Flashcard 13: Find the second derivative of $f(x) = \frac{1}{3}x^3$ .

Answer: $f''(x) = 2x$ . First derivative is $x^2$ , second derivative is $2x$ .

Flashcard 14: Calculate the second derivative of $f(x) = \frac{1}{3}x^3 + 2x$ .

Answer: $f''(x) = 2x$ . First derivative is $x^2 + 2$ , second derivative is $2x$ .

Flashcard 15: Find the third derivative of $f(x) = 5x^4 + 3x^2$ .

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

Flashcard 16: Find the second derivative of $f(x) = x^3 - 3x^2 + 3x - 1$ .

Answer: $f''(x) = 6x - 6$ . First derivative is $3x^2 - 6x + 3$ , second derivative is $6x - 6$ .

Flashcard 17: State the second derivative of $f(x) = \frac{1}{2}x^2 + 3x + 5$ .

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

Flashcard 18: State the second derivative of $f(x) = \frac{1}{2}x^2 + 3x + 5$ .

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

Flashcard 19: State the formula for the $n$ -th derivative of $f(x) = x^n$ .

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

Flashcard 20: What is the second derivative of $f(x) = \frac{1}{5}x^5$ ?

Answer: $f''(x) = 4x^3$ . Apply power rule twice: $f'(x) = x^4$ , then $f''(x) = 4x^3$ .

Flashcard 21: Calculate the third derivative of $f(x) = \frac{1}{6}x^6$ .

Answer: $f^{(3)}(x) = 20x^3$ . Apply power rule three times: $f'(x) = x^5$ , $f''(x) = 5x^4$ , $f'''(x) = 20x^3$ .

Flashcard 22: Calculate the second derivative of $f(x) = 4x^4 - 2x^2$ .

Answer: $f''(x) = 48x^2 - 4$ . Apply power rule twice: $f'(x) = 16x^3 - 4x$ , then $f''(x) = 48x^2 - 4$ .

Flashcard 23: Find the second derivative of $f(x) = \frac{1}{3}x^3$ .

Answer: $f''(x) = 2x$ . First derivative is $x^2$ , second derivative is $2x$ .

Flashcard 24: State the second derivative of $f(x) = \frac{1}{3}x^3 - x^2 + 1$ .

Answer: $f''(x) = 2x - 2$ . First derivative is $x^2 - 2x$ , second derivative is $2x - 2$ .

Flashcard 25: What is the second derivative of $f(x) = e^x$ ?

Answer: $f''(x) = e^x$ . The exponential function $e^x$ is its own derivative.

Flashcard 26: Identify the second derivative of $f(x) = \frac{1}{2}x^2 - x + 1$ .

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

Flashcard 27: What is the third derivative of $f(x) = 7x^5 - 3x^4 + x^3$ ?

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

Flashcard 28: What is the second derivative of $f(x) = e^x$ ?

Answer: $f''(x) = e^x$ . The exponential function $e^x$ is its own derivative.

Flashcard 29: What is the third derivative of $f(x) = \frac{1}{3}x^3+x^2$ ?

Answer: $f^{(3)}(x) = 2$ . First derivative is $x^2 + 2x$ , second is $2x + 2$ , third is $2$ .

Flashcard 30: Find the second derivative of $f(x) = x^5 + x^4 + x^3$ .

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

Flashcard 31: What is the second derivative of $f(x) = \frac{1}{2}x^2$ ?

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

Flashcard 32: What is the third derivative of $f(x) = 7x^5 - 3x^4 + x^3$ ?

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

Flashcard 33: Calculate the second derivative of $f(x) = 3x^3 - x^2 + x$ .

Answer: $f''(x) = 18x - 2$ . First derivative is $9x^2 - 2x + 1$ , second derivative is $18x - 2$ .

Flashcard 34: What is the second derivative of $f(x) = x^2 + 2x + 1$ ?

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

Flashcard 35: Calculate the third derivative of $f(x) = \frac{1}{6}x^6$ .

Answer: $f^{(3)}(x) = 20x^3$ . Apply power rule three times: $f'(x) = x^5$ , $f''(x) = 5x^4$ , $f'''(x) = 20x^3$ .

Flashcard 36: State the second derivative of $f(x) = \frac{1}{3}x^3 - x^2 + 1$ .

Answer: $f''(x) = 2x - 2$ . First derivative is $x^2 - 2x$ , second derivative is $2x - 2$ .

Flashcard 37: State the second derivative of $f(x) = \frac{1}{4}x^4 - x^2$ .

Answer: $f''(x) = 3x^2 - 2$ . Apply power rule twice: $f'(x) = x^3 - 2x$ , then $f''(x) = 3x^2 - 2$ .

Flashcard 38: Find the third derivative of $f(x) = 5x^4 + 3x^2$ .

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

Flashcard 39: Find the second derivative of $f(x) = x^5 + x^4 + x^3$ .

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

Flashcard 40: Find the second derivative of $f(x) = e^{3x}$ .

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

Flashcard 41: What is the second derivative of $f(x) = x^2 - 4x + 4$ ?

Answer: $f''(x) = 2$ . First derivative is $2x - 4$ , second derivative is constant $2$ .

Flashcard 42: Calculate the second derivative of $f(x) = \frac{1}{4}x^4 + x$ .

Answer: $f''(x) = 3x^2$ . First derivative is $x^3 + 1$ , second derivative is $3x^2$ .

Flashcard 43: State the second derivative of $f(x) = \frac{1}{4}x^4 - x^2$ .

Answer: $f''(x) = 3x^2 - 2$ . Apply power rule twice: $f'(x) = x^3 - 2x$ , then $f''(x) = 3x^2 - 2$ .

Flashcard 44: What is the second derivative of $f(x) = \frac{1}{5}x^5$ ?

Answer: $f''(x) = 4x^3$ . Apply power rule twice: $f'(x) = x^4$ , then $f''(x) = 4x^3$ .

Flashcard 45: Calculate the second derivative of $f(x) = 3x^3 - 3x^2 + 3x - 1$ .

Answer: $f''(x) = 18x - 6$ . First derivative is $9x^2 - 6x + 3$ , second derivative is $18x - 6$ .

Flashcard 46: What is the second derivative of $f(x) = x^3 + 3x^2 + 3x + 1$ ?

Answer: $f''(x) = 6x + 6$ . First derivative is $3x^2 + 6x + 3$ , second derivative is $6x + 6$ .

Flashcard 47: Find the second derivative of $f(x) = e^{2x}$ .

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

Flashcard 48: What is the second derivative of $f(x) = \frac{1}{2}x^2$ ?

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

Flashcard 49: Find the second derivative of $f(x) = e^{3x}$ .

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

Flashcard 50: Calculate the second derivative of $f(x) = 4x^4 - 2x^2$ .

Answer: $f''(x) = 48x^2 - 4$ . Apply power rule twice: $f'(x) = 16x^3 - 4x$ , then $f''(x) = 48x^2 - 4$ .

Flashcard 51: Calculate the second derivative of $f(x) = 3x^3 - x^2 + x$ .

Answer: $f''(x) = 18x - 2$ . First derivative is $9x^2 - 2x + 1$ , second derivative is $18x - 2$ .

Flashcard 52: What is the second derivative of $f(x) = x^3 + 3x^2 + 3x + 1$ ?

Answer: $f''(x) = 6x + 6$ . First derivative is $3x^2 + 6x + 3$ , second derivative is $6x + 6$ .

Flashcard 53: Identify the second derivative of $f(x) = \frac{1}{2}x^2 - x + 1$ .

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

Flashcard 54: What is the second derivative of $f(x) = x^4 - 2x^2 + 1$ ?

Answer: $f''(x) = 12x^2 - 4$ . First derivative is $4x^3 - 4x$ , second derivative is $12x^2 - 4$ .

Flashcard 55: State the formula for the $n$ -th derivative of $f(x) = x^n$ .

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

Flashcard 56: What is the second derivative of $f(x) = \frac{1}{5}x^5 - \frac{1}{3}x^3$ ?

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

Flashcard 57: Calculate the second derivative of $f(x) = \frac{1}{3}x^3 + 2x$ .

Answer: $f''(x) = 2x$ . First derivative is $x^2 + 2$ , second derivative is $2x$ .

Flashcard 58: Calculate the second derivative of $f(x) = 3x^3 - 3x^2 + 3x - 1$ .

Answer: $f''(x) = 18x - 6$ . First derivative is $9x^2 - 6x + 3$ , second derivative is $18x - 6$ .

Flashcard 59: What is the second derivative of $f(x) = x^2 - 4x + 4$ ?

Answer: $f''(x) = 2$ . First derivative is $2x - 4$ , second derivative is constant $2$ .

Flashcard 60: What is the second derivative of $f(x) = \frac{1}{5}x^5 - \frac{1}{3}x^3$ ?

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

Flashcard 61: Calculate the second derivative of $f(x) = \frac{1}{x}$ .

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

AP Calculus AB Flashcards: Calculating Higher Order Derivatives

Calculating Higher Order Derivatives

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: What is the second derivative of f(x)=x3f(x) = x^3f(x)=x3?

Flashcard 2: Calculate the third derivative of f(x)=x5−x3+xf(x) = x^5 - x^3 + xf(x)=x5−x3+x.

Flashcard 3: Calculate the second derivative of f(x)=14x4+xf(x) = \frac{1}{4}x^4 + xf(x)=41​x4+x.

Flashcard 4: State the second derivative of f(x)=12x2−xf(x) = \frac{1}{2}x^2 - xf(x)=21​x2−x.

Flashcard 5: What is the second derivative of f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1?

Flashcard 6: Find the second derivative of f(x)=e2xf(x) = e^{2x}f(x)=e2x.

Flashcard 7: State the second derivative of f(x)=12x2−xf(x) = \frac{1}{2}x^2 - xf(x)=21​x2−x.

Flashcard 8: Find the second derivative of f(x)=x3−3x2+3x−1f(x) = x^3 - 3x^2 + 3x - 1f(x)=x3−3x2+3x−1.

Flashcard 9: What is the third derivative of f(x)=13x3+x2f(x) = \frac{1}{3}x^3+x^2f(x)=31​x3+x2?

Flashcard 10: Calculate the third derivative of f(x)=x5−x3+xf(x) = x^5 - x^3 + xf(x)=x5−x3+x.

Flashcard 11: What is the second derivative of f(x)=x4−2x2+1f(x) = x^4 - 2x^2 + 1f(x)=x4−2x2+1?

Flashcard 12: Calculate the second derivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​.

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

Flashcard 14: Calculate the second derivative of f(x)=13x3+2xf(x) = \frac{1}{3}x^3 + 2xf(x)=31​x3+2x.

Flashcard 15: Find the third derivative of f(x)=5x4+3x2f(x) = 5x^4 + 3x^2f(x)=5x4+3x2.

Flashcard 16: Find the second derivative of f(x)=x3−3x2+3x−1f(x) = x^3 - 3x^2 + 3x - 1f(x)=x3−3x2+3x−1.

Flashcard 17: State the second derivative of f(x)=12x2+3x+5f(x) = \frac{1}{2}x^2 + 3x + 5f(x)=21​x2+3x+5.

Flashcard 18: State the second derivative of f(x)=12x2+3x+5f(x) = \frac{1}{2}x^2 + 3x + 5f(x)=21​x2+3x+5.

Flashcard 19: State the formula for the nnn-th derivative of f(x)=xnf(x) = x^nf(x)=xn.

Flashcard 20: What is the second derivative of f(x)=15x5f(x) = \frac{1}{5}x^5f(x)=51​x5?

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

Flashcard 22: Calculate the second derivative of f(x)=4x4−2x2f(x) = 4x^4 - 2x^2f(x)=4x4−2x2.

Flashcard 23: Find the second derivative of f(x)=13x3f(x) = \frac{1}{3}x^3f(x)=31​x3.

Flashcard 24: State the second derivative of f(x)=13x3−x2+1f(x) = \frac{1}{3}x^3 - x^2 + 1f(x)=31​x3−x2+1.

Flashcard 25: What is the second derivative of f(x)=exf(x) = e^xf(x)=ex?

Flashcard 26: Identify the second derivative of f(x)=12x2−x+1f(x) = \frac{1}{2}x^2 - x + 1f(x)=21​x2−x+1.

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

Flashcard 28: What is the second derivative of f(x)=exf(x) = e^xf(x)=ex?

Flashcard 29: What is the third derivative of f(x)=13x3+x2f(x) = \frac{1}{3}x^3+x^2f(x)=31​x3+x2?

Flashcard 30: Find the second derivative of f(x)=x5+x4+x3f(x) = x^5 + x^4 + x^3f(x)=x5+x4+x3.

Flashcard 31: What is the second derivative of f(x)=12x2f(x) = \frac{1}{2}x^2f(x)=21​x2?

Flashcard 32: What is the third derivative of f(x)=7x5−3x4+x3f(x) = 7x^5 - 3x^4 + x^3f(x)=7x5−3x4+x3?

Flashcard 33: Calculate the second derivative of f(x)=3x3−x2+xf(x) = 3x^3 - x^2 + xf(x)=3x3−x2+x.

Flashcard 34: What is the second derivative of f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1?

Flashcard 35: Calculate the third derivative of f(x)=16x6f(x) = \frac{1}{6}x^6f(x)=61​x6.

Flashcard 36: State the second derivative of f(x)=13x3−x2+1f(x) = \frac{1}{3}x^3 - x^2 + 1f(x)=31​x3−x2+1.

Flashcard 37: State the second derivative of f(x)=14x4−x2f(x) = \frac{1}{4}x^4 - x^2f(x)=41​x4−x2.

Flashcard 38: Find the third derivative of f(x)=5x4+3x2f(x) = 5x^4 + 3x^2f(x)=5x4+3x2.

Flashcard 39: Find the second derivative of f(x)=x5+x4+x3f(x) = x^5 + x^4 + x^3f(x)=x5+x4+x3.

Flashcard 40: Find the second derivative of f(x)=e3xf(x) = e^{3x}f(x)=e3x.

Flashcard 41: What is the second derivative of f(x)=x2−4x+4f(x) = x^2 - 4x + 4f(x)=x2−4x+4?

Flashcard 42: Calculate the second derivative of f(x)=14x4+xf(x) = \frac{1}{4}x^4 + xf(x)=41​x4+x.

Flashcard 43: State the second derivative of f(x)=14x4−x2f(x) = \frac{1}{4}x^4 - x^2f(x)=41​x4−x2.

Flashcard 44: What is the second derivative of f(x)=15x5f(x) = \frac{1}{5}x^5f(x)=51​x5?

Flashcard 45: Calculate the second derivative of f(x)=3x3−3x2+3x−1f(x) = 3x^3 - 3x^2 + 3x - 1f(x)=3x3−3x2+3x−1.

Flashcard 46: What is the second derivative of f(x)=x3+3x2+3x+1f(x) = x^3 + 3x^2 + 3x + 1f(x)=x3+3x2+3x+1?

Flashcard 47: Find the second derivative of f(x)=e2xf(x) = e^{2x}f(x)=e2x.

Flashcard 48: What is the second derivative of f(x)=12x2f(x) = \frac{1}{2}x^2f(x)=21​x2?

Flashcard 49: Find the second derivative of f(x)=e3xf(x) = e^{3x}f(x)=e3x.

Flashcard 50: Calculate the second derivative of f(x)=4x4−2x2f(x) = 4x^4 - 2x^2f(x)=4x4−2x2.

Flashcard 51: Calculate the second derivative of f(x)=3x3−x2+xf(x) = 3x^3 - x^2 + xf(x)=3x3−x2+x.

Flashcard 52: What is the second derivative of f(x)=x3+3x2+3x+1f(x) = x^3 + 3x^2 + 3x + 1f(x)=x3+3x2+3x+1?

Flashcard 53: Identify the second derivative of f(x)=12x2−x+1f(x) = \frac{1}{2}x^2 - x + 1f(x)=21​x2−x+1.

Flashcard 54: What is the second derivative of f(x)=x4−2x2+1f(x) = x^4 - 2x^2 + 1f(x)=x4−2x2+1?

Flashcard 55: State the formula for the nnn-th derivative of f(x)=xnf(x) = x^nf(x)=xn.

Flashcard 56: 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?

Flashcard 57: Calculate the second derivative of f(x)=13x3+2xf(x) = \frac{1}{3}x^3 + 2xf(x)=31​x3+2x.

Flashcard 58: Calculate the second derivative of f(x)=3x3−3x2+3x−1f(x) = 3x^3 - 3x^2 + 3x - 1f(x)=3x3−3x2+3x−1.

Flashcard 59: What is the second derivative of f(x)=x2−4x+4f(x) = x^2 - 4x + 4f(x)=x2−4x+4?

Flashcard 60: 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?

Flashcard 61: Calculate the second derivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​.

Flashcard 1: What is the second derivative of $f(x) = x^3$ ?

Flashcard 2: Calculate the third derivative of $f(x) = x^5 - x^3 + x$ .

Flashcard 3: Calculate the second derivative of $f(x) = \frac{1}{4}x^4 + x$ .

Flashcard 4: State the second derivative of $f(x) = \frac{1}{2}x^2 - x$ .

Flashcard 5: What is the second derivative of $f(x) = x^2 + 2x + 1$ ?

Flashcard 6: Find the second derivative of $f(x) = e^{2x}$ .

Flashcard 7: State the second derivative of $f(x) = \frac{1}{2}x^2 - x$ .

Flashcard 8: Find the second derivative of $f(x) = x^3 - 3x^2 + 3x - 1$ .

Flashcard 9: What is the third derivative of $f(x) = \frac{1}{3}x^3+x^2$ ?

Flashcard 10: Calculate the third derivative of $f(x) = x^5 - x^3 + x$ .

Flashcard 11: What is the second derivative of $f(x) = x^4 - 2x^2 + 1$ ?

Flashcard 12: Calculate the second derivative of $f(x) = \frac{1}{x}$ .

Flashcard 13: Find the second derivative of $f(x) = \frac{1}{3}x^3$ .

Flashcard 14: Calculate the second derivative of $f(x) = \frac{1}{3}x^3 + 2x$ .

Flashcard 15: Find the third derivative of $f(x) = 5x^4 + 3x^2$ .

Flashcard 16: Find the second derivative of $f(x) = x^3 - 3x^2 + 3x - 1$ .

Flashcard 17: State the second derivative of $f(x) = \frac{1}{2}x^2 + 3x + 5$ .

Flashcard 18: State the second derivative of $f(x) = \frac{1}{2}x^2 + 3x + 5$ .

Flashcard 19: State the formula for the $n$ -th derivative of $f(x) = x^n$ .

Flashcard 20: What is the second derivative of $f(x) = \frac{1}{5}x^5$ ?

Flashcard 21: Calculate the third derivative of $f(x) = \frac{1}{6}x^6$ .

Flashcard 22: Calculate the second derivative of $f(x) = 4x^4 - 2x^2$ .

Flashcard 23: Find the second derivative of $f(x) = \frac{1}{3}x^3$ .

Flashcard 24: State the second derivative of $f(x) = \frac{1}{3}x^3 - x^2 + 1$ .

Flashcard 25: What is the second derivative of $f(x) = e^x$ ?

Flashcard 26: Identify the second derivative of $f(x) = \frac{1}{2}x^2 - x + 1$ .

Flashcard 27: What is the third derivative of $f(x) = 7x^5 - 3x^4 + x^3$ ?

Flashcard 28: What is the second derivative of $f(x) = e^x$ ?

Flashcard 29: What is the third derivative of $f(x) = \frac{1}{3}x^3+x^2$ ?

Flashcard 30: Find the second derivative of $f(x) = x^5 + x^4 + x^3$ .

Flashcard 31: What is the second derivative of $f(x) = \frac{1}{2}x^2$ ?

Flashcard 32: What is the third derivative of $f(x) = 7x^5 - 3x^4 + x^3$ ?

Flashcard 33: Calculate the second derivative of $f(x) = 3x^3 - x^2 + x$ .

Flashcard 34: What is the second derivative of $f(x) = x^2 + 2x + 1$ ?

Flashcard 35: Calculate the third derivative of $f(x) = \frac{1}{6}x^6$ .

Flashcard 36: State the second derivative of $f(x) = \frac{1}{3}x^3 - x^2 + 1$ .

Flashcard 37: State the second derivative of $f(x) = \frac{1}{4}x^4 - x^2$ .

Flashcard 38: Find the third derivative of $f(x) = 5x^4 + 3x^2$ .

Flashcard 39: Find the second derivative of $f(x) = x^5 + x^4 + x^3$ .

Flashcard 40: Find the second derivative of $f(x) = e^{3x}$ .

Flashcard 41: What is the second derivative of $f(x) = x^2 - 4x + 4$ ?

Flashcard 42: Calculate the second derivative of $f(x) = \frac{1}{4}x^4 + x$ .

Flashcard 43: State the second derivative of $f(x) = \frac{1}{4}x^4 - x^2$ .

Flashcard 44: What is the second derivative of $f(x) = \frac{1}{5}x^5$ ?

Flashcard 45: Calculate the second derivative of $f(x) = 3x^3 - 3x^2 + 3x - 1$ .

Flashcard 46: What is the second derivative of $f(x) = x^3 + 3x^2 + 3x + 1$ ?

Flashcard 47: Find the second derivative of $f(x) = e^{2x}$ .

Flashcard 48: What is the second derivative of $f(x) = \frac{1}{2}x^2$ ?

Flashcard 49: Find the second derivative of $f(x) = e^{3x}$ .

Flashcard 50: Calculate the second derivative of $f(x) = 4x^4 - 2x^2$ .

Flashcard 51: Calculate the second derivative of $f(x) = 3x^3 - x^2 + x$ .

Flashcard 52: What is the second derivative of $f(x) = x^3 + 3x^2 + 3x + 1$ ?

Flashcard 53: Identify the second derivative of $f(x) = \frac{1}{2}x^2 - x + 1$ .

Flashcard 54: What is the second derivative of $f(x) = x^4 - 2x^2 + 1$ ?

Flashcard 55: State the formula for the $n$ -th derivative of $f(x) = x^n$ .

Flashcard 56: What is the second derivative of $f(x) = \frac{1}{5}x^5 - \frac{1}{3}x^3$ ?

Flashcard 57: Calculate the second derivative of $f(x) = \frac{1}{3}x^3 + 2x$ .

Flashcard 58: Calculate the second derivative of $f(x) = 3x^3 - 3x^2 + 3x - 1$ .

Flashcard 59: What is the second derivative of $f(x) = x^2 - 4x + 4$ ?

Flashcard 60: What is the second derivative of $f(x) = \frac{1}{5}x^5 - \frac{1}{3}x^3$ ?

Flashcard 61: Calculate the second derivative of $f(x) = \frac{1}{x}$ .