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AP Calculus AB Flashcards: Derivatives Of Trigonometry And Logarithmic Functions

Study Derivatives Of Trigonometry And Logarithmic Functions 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 Derivatives Of Trigonometry And Logarithmic Functions, 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: Derivatives Of Trigonometry And Logarithmic Functions

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QUESTION

What is the derivative of f(x)=sin⁡2(x)f(x) = \sin^2(x)f(x)=sin2(x)?

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ANSWER

f′(x)=2sin⁡(x)cos⁡(x)f'(x) = 2\sin(x)\cos(x)f′(x)=2sin(x)cos(x). Apply chain rule to the squared function.

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Flashcard 1: What is the derivative of f(x)=sin⁡2(x)f(x) = \sin^2(x)f(x)=sin2(x)?

Answer: f′(x)=2sin⁡(x)cos⁡(x)f'(x) = 2\sin(x)\cos(x)f′(x)=2sin(x)cos(x). Apply chain rule to the squared function.

Flashcard 2: Determine the derivative of f(x)=3sin(x)f(x) = 3\text{sin}(x)f(x)=3sin(x).

Answer: f′(x)=3cos(x)f'(x) = 3\text{cos}(x)f′(x)=3cos(x). Use the constant multiple rule: multiply the derivative by 3.

Flashcard 3: What is the second derivative of sin(x)\text{sin}(x)sin(x)?

Answer: −sin(x)-\text{sin}(x)−sin(x). Take the derivative of cos(x)\text{cos}(x)cos(x) to get the second derivative.

Flashcard 4: What is the second derivative of cos(x)\text{cos}(x)cos(x)?

Answer: −cos(x)-\text{cos}(x)−cos(x). Take the derivative of −sin(x)-\text{sin}(x)−sin(x) to get the second derivative.

Flashcard 5: What is the second derivative of ln(x)\text{ln}(x)ln(x)?

Answer: −1x2-\frac{1}{x^2}−x21​. Take the derivative of 1x\frac{1}{x}x1​ to get the second derivative.

Flashcard 6: Evaluate the derivative at x=π2x = \frac{\text{π}}{2}x=2π​: f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x).

Answer: f′(π2)=0f'(\frac{\text{π}}{2}) = 0f′(2π​)=0. The derivative is cos(x)\text{cos}(x)cos(x), and cos(π2)=0\text{cos}(\frac{\pi}{2}) = 0cos(2π​)=0.

Flashcard 7: What is the derivative of f(x)=2ln(x)f(x) = 2\text{ln}(x)f(x)=2ln(x)?

Answer: f′(x)=2xf'(x) = \frac{2}{x}f′(x)=x2​. Use the constant multiple rule with coefficient 2.

Flashcard 8: Determine the derivative of f(x)=4exf(x) = 4e^xf(x)=4ex.

Answer: f′(x)=4exf'(x) = 4e^xf′(x)=4ex. Use the constant multiple rule: multiply the derivative by 4.

Flashcard 9: What is the derivative of cos(x)\text{cos}(x)cos(x)?

Answer: −sin(x)-\text{sin}(x)−sin(x). The derivative of cosine is negative sine.

Flashcard 10: What is the derivative of f(x)=cos(x)−exf(x) = \text{cos}(x) - e^xf(x)=cos(x)−ex?

Answer: f′(x)=−sin(x)−exf'(x) = -\text{sin}(x) - e^xf′(x)=−sin(x)−ex. Apply linearity: derivative of difference equals difference of derivatives.

Flashcard 11: Evaluate the derivative at x=0x = 0x=0: f(x)=cos⁡(x)f(x) = \cos(x)f(x)=cos(x).

Answer: f′(0)=0f'(0) = 0f′(0)=0. The derivative is −sin⁡(x)-\sin(x)−sin(x), and −sin⁡(0)=0-\sin(0) = 0−sin(0)=0.

Flashcard 12: What is the derivative of sin(x)\text{sin}(x)sin(x)?

Answer: cos(x)\text{cos}(x)cos(x). The derivative of sine is cosine.

Flashcard 13: What is the derivative of f(x)=cos(x)×ln(x)f(x) = \text{cos}(x) \times \text{ln}(x)f(x)=cos(x)×ln(x)?

Answer: f′(x)=−sin(x)ln(x)+cos(x)xf'(x) = -\text{sin}(x)\text{ln}(x) + \frac{\text{cos}(x)}{x}f′(x)=−sin(x)ln(x)+xcos(x)​. Apply the product rule to both functions.

Flashcard 14: What is the derivative of exe^xex?

Answer: exe^xex. The exponential function is its own derivative.

Flashcard 15: Evaluate the derivative of f(x)=ln(x)f(x) = \text{ln}(x)f(x)=ln(x) at x=1x = 1x=1.

Answer: f′(1)=1f'(1) = 1f′(1)=1. The derivative of ln(x)\text{ln}(x)ln(x) is 1x\frac{1}{x}x1​, and 11=1\frac{1}{1} = 111​=1.

Flashcard 16: Find the derivative of f(x)=ex+ln(x)f(x) = e^x + \text{ln}(x)f(x)=ex+ln(x).

Answer: f′(x)=ex+1xf'(x) = e^x + \frac{1}{x}f′(x)=ex+x1​. Apply linearity to find the sum of individual derivatives.

Flashcard 17: What is the derivative of f(x)=sin(x)×exf(x) = \text{sin}(x) \times e^xf(x)=sin(x)×ex?

Answer: f′(x)=cos(x)ex+sin(x)exf'(x) = \text{cos}(x)e^x + \text{sin}(x)e^xf′(x)=cos(x)ex+sin(x)ex. Apply the product rule: (uv)′=u′v+uv′(uv)' = u'v + uv'(uv)′=u′v+uv′.

Flashcard 18: Evaluate the derivative at x=0x = 0x=0: f(x)=ln⁡(x+1)f(x) = \ln(x + 1)f(x)=ln(x+1).

Answer: f′(0)=1f'(0) = 1f′(0)=1. Use chain rule: derivative is 1x+1\frac{1}{x+1}x+11​, and 10+1=1\frac{1}{0+1} = 10+11​=1.

Flashcard 19: Find the third derivative of f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x).

Answer: −cos(x)-\text{cos}(x)−cos(x). The third derivative follows the sine pattern: sin → cos → -sin → -cos.

Flashcard 20: What is the derivative of ln(x)\text{ln}(x)ln(x)?

Answer: 1x\frac{1}{x}x1​. The derivative of natural logarithm is the reciprocal.

Flashcard 21: What is the derivative of f(x)=sin(x)+exf(x) = \text{sin}(x) + e^xf(x)=sin(x)+ex?

Answer: f′(x)=cos(x)+exf'(x) = \text{cos}(x) + e^xf′(x)=cos(x)+ex. Apply linearity to find the sum of derivatives.

Flashcard 22: Find the derivative of f(x)=ex2f(x) = e^{x^2}f(x)=ex2 using chain rule.

Answer: f′(x)=2xex2f'(x) = 2xe^{x^2}f′(x)=2xex2. Apply chain rule with outer function eue^ueu and inner x2x^2x2.

Flashcard 23: Find the third derivative of f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x).

Answer: sin(x)\text{sin}(x)sin(x). The third derivative follows the cosine pattern: cos → -sin → -cos → sin.

Flashcard 24: Evaluate the derivative at x=0x = 0x=0: f(x)=exf(x) = e^xf(x)=ex.

Answer: f′(0)=1f'(0) = 1f′(0)=1. The derivative of exe^xex is exe^xex, and e0=1e^0 = 1e0=1.

Flashcard 25: What is the derivative of f(x)=cos2(x)f(x) = \text{cos}^2(x)f(x)=cos2(x)?

Answer: f′(x)=−2sin(x)cos(x)f'(x) = -2\text{sin}(x)\text{cos}(x)f′(x)=−2sin(x)cos(x). Apply chain rule to the squared function.

Flashcard 26: Find the derivative of f(x)=ln(x2)f(x) = \text{ln}(x^2)f(x)=ln(x2).

Answer: f′(x)=2xf'(x) = \frac{2}{x}f′(x)=x2​. Use logarithm property: ln(x2)=2ln(x)\text{ln}(x^2) = 2\text{ln}(x)ln(x2)=2ln(x).

Flashcard 27: What is the derivative of f(x)=sin(2x)f(x) = \text{sin}(2x)f(x)=sin(2x)?

Answer: f′(x)=2cos(2x)f'(x) = 2\text{cos}(2x)f′(x)=2cos(2x). Apply chain rule: outer derivative times inner derivative.

Flashcard 28: What is the second derivative of exe^xex?

Answer: exe^xex. The exponential function equals its own second derivative.

Flashcard 29: What is the derivative of f(x)=cos(x)+ln(x)f(x) = \text{cos}(x) + \text{ln}(x)f(x)=cos(x)+ln(x)?

Answer: f′(x)=−sin(x)+1xf'(x) = -\text{sin}(x) + \frac{1}{x}f′(x)=−sin(x)+x1​. Apply linearity to find the sum of derivatives.

Flashcard 30: Find the derivative of f(x)=−cos(x)f(x) = -\text{cos}(x)f(x)=−cos(x).

Answer: f′(x)=sin(x)f'(x) = \text{sin}(x)f′(x)=sin(x). Factor out the negative sign from the derivative.