AP Calculus AB Flashcards: Derivatives Of Trigonometry And Logarithmic Functions

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.

Flashcard 1: What is the derivative of $f(x) = \text{sin}(2x)$?

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

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

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

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

Answer: $-\frac{1}{x^2}$. Take the derivative of $\frac{1}{x}$ to get the second derivative.

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

Answer: $-\frac{1}{x^2}$. Take the derivative of $\frac{1}{x}$ to get the second derivative.

Flashcard 5: Find the derivative of $f(x) = -\text{cos}(x)$.

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

Flashcard 6: What is the derivative of $\text{sin}(x)$?

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

Flashcard 7: What is the second derivative of $\text{sin}(x)$?

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

Flashcard 8: Find the derivative of $f(x) = e^x \times \text{ln}(x)$.

Answer: $f'(x) = e^x \times \frac{1}{x} + \text{ln}(x) \times e^x$. Use the product rule on $e^x$ and $\text{ln}(x)$.

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

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

Flashcard 10: What is the derivative of $f(x) = \text{cos}(x) \times \text{ln}(x)$?

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

Flashcard 11: What is the derivative of $e^x$?

Answer: $e^x$. The exponential function is its own derivative.

Flashcard 12: Evaluate the derivative of $f(x) = \text{ln}(x)$ at $x = 1$.

Answer: $f'(1) = 1$. The derivative of $\text{ln}(x)$ is $\frac{1}{x}$, and $\frac{1}{1} = 1$.

Flashcard 13: Find the derivative of $f(x) = e^x + \text{ln}(x)$.

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

Flashcard 14: What is the derivative of $\text{ln}(x)$?

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

Flashcard 15: Find the derivative of $f(x) = \text{ln}(x^2)$.

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

Flashcard 16: What is the derivative of $\text{cos}(x)$?

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

Flashcard 17: Evaluate the derivative at $x = \frac{\text{π}}{2}$: $f(x) = \text{sin}(x)$.

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

Flashcard 18: What is the derivative of $f(x) = \text{cos}(x) - e^x$?

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

Flashcard 19: What is the derivative of $\text{cos}(x)$?

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

Flashcard 20: Find the derivative of $f(x) = e^{x^2}$ using chain rule.

Answer: $f'(x) = 2xe^{x^2}$. Apply chain rule with outer function $e^u$ and inner $x^2$.

Flashcard 21: What is the second derivative of $e^x$?

Answer: $e^x$. The exponential function equals its own second derivative.

Flashcard 22: Evaluate the derivative at $x = 0$: $f(x) = \text{ln}(x + 1)$.

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

Flashcard 23: Evaluate the derivative at $x = \frac{\text{π}}{2}$: $f(x) = \text{sin}(x)$.

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

Flashcard 24: What is the second derivative of $\text{cos}(x)$?

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

Flashcard 25: Find the derivative of $f(x) = -\text{sin}(x)$.

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

Flashcard 26: What is the derivative of $f(x) = \text{sin}(x) + e^x$?

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

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

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

Flashcard 28: Evaluate the derivative at $x = 0$: $f(x) = \text{cos}(x)$.

Answer: $f'(0) = 0$. The derivative is $-\text{sin}(x)$, and $-\text{sin}(0) = 0$.

Flashcard 29: What is the derivative of $f(x) = 2\text{ln}(x)$?

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

Flashcard 30: Evaluate the derivative at $x = 0$: $f(x) = e^x$.

Answer: $f'(0) = 1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

Flashcard 31: What is the second derivative of $\text{sin}(x)$?

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

Flashcard 32: Find the third derivative of $f(x) = \text{cos}(x)$.

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

Flashcard 33: Find the derivative of $f(x) = \text{ln}(2x)$.

Answer: $f'(x) = \frac{1}{x}$. The derivative simplifies using logarithm properties.

Flashcard 34: Evaluate the derivative at $x = 0$: $f(x) = \ln(x + 1)$.

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

Flashcard 35: Find the third derivative of $f(x) = \text{sin}(x)$.

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

Flashcard 36: What is the derivative of $f(x) = \text{sin}(x) \times e^x$?

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

Flashcard 37: Determine the derivative of $f(x) = 3\text{sin}(x)$.

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

Flashcard 38: Find the derivative of $f(x) = e^x \times \text{ln}(x)$.

Answer: $f'(x) = e^x \times \frac{1}{x} + \text{ln}(x) \times e^x$. Use the product rule on $e^x$ and $\text{ln}(x)$.

Flashcard 39: What is the derivative of $\text{ln}(x)$?

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

Flashcard 40: What is the derivative of $f(x) = \text{cos}(2x)$?

Answer: $f'(x) = -2\text{sin}(2x)$. Apply chain rule with inner function $2x$.

Flashcard 41: What is the derivative of $f(x) = \text{cos}(x) - e^x$?

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

Flashcard 42: Find the derivative of $f(x) = \text{ln}(2x)$.

Answer: $f'(x) = \frac{1}{x}$. The derivative simplifies using logarithm properties.

Flashcard 43: What is the derivative of $f(x) = \text{sin}(x) + e^x$?

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

Flashcard 44: What is the derivative of $f(x) = \sin^2(x)$?

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

Flashcard 45: Evaluate the derivative of $f(x) = \text{ln}(x)$ at $x = 1$.

Answer: $f'(1) = 1$. The derivative of $\text{ln}(x)$ is $\frac{1}{x}$, and $\frac{1}{1} = 1$.

Flashcard 46: What is the derivative of $f(x) = \text{sin}(2x)$?

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

Flashcard 47: Find the third derivative of $f(x) = \text{cos}(x)$.

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

Flashcard 48: What is the derivative of $f(x) = \text{cos}(2x)$?

Answer: $f'(x) = -2\text{sin}(2x)$. Apply chain rule with inner function $2x$.

Flashcard 49: What is the derivative of $f(x) = \text{cos}^2(x)$?

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

Flashcard 50: Find the derivative of $f(x) = \text{sin}(x) + \text{cos}(x)$.

Answer: $f'(x) = \text{cos}(x) - \text{sin}(x)$. Use linearity: derivative of sum equals sum of derivatives.

Flashcard 51: Evaluate the derivative at $x = 0$: $f(x) = e^x$.

Answer: $f'(0) = 1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

Flashcard 52: Find the derivative of $f(x) = e^{x^2}$ using chain rule.

Answer: $f'(x) = 2xe^{x^2}$. Apply chain rule with outer function $e^u$ and inner $x^2$.

Flashcard 53: Find the derivative of $f(x) = e^{2x}$.

Answer: $f'(x) = 2e^{2x}$. Use chain rule: derivative of $e^u$ is $e^u \cdot u'$.

Flashcard 54: Find the derivative of $f(x) = \text{sin}(x) + \text{cos}(x)$.

Answer: $f'(x) = \text{cos}(x) - \text{sin}(x)$. Use linearity: derivative of sum equals sum of derivatives.

Flashcard 55: Determine the derivative of $f(x) = 4e^x$.

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

Flashcard 56: Find the third derivative of $f(x) = \text{sin}(x)$.

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

Flashcard 57: What is the derivative of $f(x) = \text{sin}^2(x)$?

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

Flashcard 58: Find the derivative of $f(x) = \text{ln}(x^2)$.

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

Flashcard 59: Determine the derivative of $f(x) = 3\text{sin}(x)$.

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

Flashcard 60: What is the second derivative of $e^x$?

Answer: $e^x$. The exponential function equals its own second derivative.

Flashcard 61: What is the derivative of $\text{sin}(x)$?

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

Flashcard 62: What is the derivative of $f(x) = \text{sin}(x) \times e^x$?

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

Flashcard 63: Determine the derivative of $f(x) = 4e^x$.

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

Flashcard 64: Find the derivative of $f(x) = -\text{cos}(x)$.

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

Flashcard 65: What is the derivative of $f(x) = \text{cos}(x) + \text{ln}(x)$?

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

Flashcard 66: Find the derivative of $f(x) = e^x + \text{ln}(x)$.

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

Flashcard 67: Find the derivative of $f(x) = -\text{sin}(x)$.

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

Flashcard 68: What is the derivative of $f(x) = 2\text{ln}(x)$?

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

Flashcard 69: What is the derivative of $e^x$?

Answer: $e^x$. The exponential function is its own derivative.

Flashcard 70: Find the derivative of $f(x) = e^{2x}$.

Answer: $f'(x) = 2e^{2x}$. Use chain rule: derivative of $e^u$ is $e^u \cdot u'$.

Flashcard 71: Evaluate the derivative at $x = 0$: $f(x) = \cos(x)$.

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