AP Calculus AB Flashcards: The Product Rule

What this deck covers

This deck focuses on The Product Rule, 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: State the Product Rule formula for differentiation.

Answer: $(f \times g)' = f' \times g + f \times g'$. Standard formula where each function multiplies the other's derivative.

Flashcard 2: What is the outcome of differentiating $f(x) = xe^x$ using Product Rule?

Answer: $e^x + xe^x$. Product Rule: $1 \times e^x + x \times e^x$.

Flashcard 3: Find $d/dx$ of $f(x) = x^2 \times \text{sec}(x)$. Use Product Rule.

Answer: $2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.

Flashcard 4: Does the Product Rule apply to $f(x) = x^2 \times x^3$?

Answer: Yes, but it can be simplified before applying. Could simplify to $x^5$ first, but Product Rule still applies.

Flashcard 5: Which rule differentiates the product of two differentiable functions?

Answer: The Product Rule. Standard rule for finding derivatives of function products.

Flashcard 6: What is $d/dx$ for $h(x) = x^3 \times \text{ln}(x)$ using the Product Rule?

Answer: $3x^2 \text{ln}(x) + x^2$. Product Rule: $(x^3)' \times \ln(x) + x^3 \times (\ln(x))'$.

Flashcard 7: Calculate the derivative of $h(x) = x \text{ln}(x)$.

Answer: $1 \times \text{ln}(x) + \frac{x}{x}$. Product Rule: $1 \times \ln(x) + x \times \frac{1}{x}$.

Flashcard 8: What is the result of differentiating $u(x) = e^x \times \text{ln}(x)$?

Answer: $e^x \times \text{ln}(x) + \frac{e^x}{x}$. Product Rule: $(e^x)' \times \ln(x) + e^x \times (\ln(x))'$.

Flashcard 9: Identify the derivative of $y = (3x + 2)(x^3 - 1)$ using the Product Rule.

Answer: $9x^3 + 6x^2 - 3x - 2$. Product Rule: $3 \times (x^3 - 1) + (3x + 2) \times 3x^2$.

Flashcard 10: What is the derivative of $h(x) = x^2 \times \text{sec}(x)$ using Product Rule?

Answer: $2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.

Flashcard 11: Evaluate the derivative: $h(x) = x^2 e^x$ using the Product Rule.

Answer: $2x e^x + x^2 e^x$. Product Rule: $(x^2)' \times e^x + x^2 \times (e^x)'$.

Flashcard 12: Is the Product Rule applicable to $f(x) = (x^2 + x)(x - 1)$?

Answer: Yes, it is applicable. Two differentiable functions multiplied together require Product Rule.

Flashcard 13: Calculate the derivative for $y = (\text{sin}(x))(x^2)$ using Product Rule.

Answer: $2x \times \text{sin}(x) + x^2 \times \text{cos}(x)$. Product Rule applied with order switched: same result.

Flashcard 14: What is the derivative of $f(x) = (x + 2)(x^2 + 1)$?

Answer: $3x^2 + 4x + 2$. Product Rule: $1 \times (x^2 + 1) + (x + 2) \times 2x$.

Flashcard 15: Find the derivative: $f(x) = (2x + 1)(x^2 - 4x + 4)$. Use Product Rule.

Answer: $2x^3 - 6x^2 + 10x - 4$. Product Rule: $2 \times (x^2 - 4x + 4) + (2x + 1) \times (2x - 4)$.

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

Answer: $3x^2 - 2x + 2$. Product Rule: $1 \times (x^2 - x + 1) + (x + 3) \times (2x - 1)$.

Flashcard 17: What is $d/dx$ of $f(x) = (x^2)(\text{cos}(x))$ using Product Rule?

Answer: $2x \text{cos}(x) - x^2 \text{sin}(x)$. Product Rule: $(x^2)' \times \cos(x) + x^2 \times (\cos(x))'$.

Flashcard 18: Calculate the derivative: $y = (x^4)(\text{sin}(x))$ using Product Rule.

Answer: $4x^3 \text{sin}(x) + x^4 \text{cos}(x)$. Product Rule: $(x^4)' \times \sin(x) + x^4 \times (\sin(x))'$.

Flashcard 19: Find the derivative of $h(x) = x^2 \times \text{ln}(x)$.

Answer: $2x \text{ln}(x) + x$. Product Rule: $(x^2)' \times \ln(x) + x^2 \times (\ln(x))'$.

Flashcard 20: Is the Product Rule used for $f(x) = (x + 1)(x^2 - 1)$?

Answer: Yes, it is applicable. Product of two functions requires the Product Rule.

Flashcard 21: Evaluate the derivative: $y = (x^3 + 2)(x^2 - 1)$ using Product Rule.

Answer: $5x^4 - 3x^2 + 4x - 2$. Product Rule: $(3x^2) \times (x^2 - 1) + (x^3 + 2) \times 2x$.

Flashcard 22: Find the derivative of $y = (x^2 + 3x + 1)(x - 2)$.

Answer: $3x^2 + 2x - 5$. Product Rule: $(2x + 3) \times (x - 2) + (x^2 + 3x + 1) \times 1$.

Flashcard 23: Determine the derivative: $f(x) = (x^3)(\text{tan}(x))$. Use Product Rule.

Answer: $3x^2 \text{tan}(x) + x^3 \text{sec}^2(x)$. Product Rule: $(x^3)' \times \tan(x) + x^3 \times (\tan(x))'$.

Flashcard 24: Find $d/dx$ of $y = x(x^2 + 1)$ using Product Rule.

Answer: $3x^2 + 1$. Product Rule: $1 \times (x^2 + 1) + x \times 2x$.

Flashcard 25: Determine the derivative: $y = (x^5)(\text{cos}(x))$. Use Product Rule.

Answer: $5x^4 \text{cos}(x) - x^5 \text{sin}(x)$. Product Rule: $(x^5)' \times \cos(x) + x^5 \times (\cos(x))'$.

Flashcard 26: What is the derivative of $y = (x^4)(e^x)$ using Product Rule?

Answer: $4x^3 e^x + x^4 e^x$. Product Rule: $(x^4)' \times e^x + x^4 \times (e^x)'$.

Flashcard 27: Which rule is used to differentiate the product of two functions?

Answer: The Product Rule. Essential rule for differentiating products of functions.

Flashcard 28: Find the derivative: $y = (3x^2 + x)(x^2 - x)$ using Product Rule.

Answer: $9x^3 - 4x^2 + x$. Product Rule: $(6x + 1) \times (x^2 - x) + (3x^2 + x) \times (2x - 1)$.

Flashcard 29: Which component is found first in the Product Rule: $f'(x)$ or $g'(x)$?

Answer: Either order is acceptable. The Product Rule is symmetric; order doesn't matter.

Flashcard 30: Find the derivative of $f(x) = (x^2 + 1)(\text{sin}(x))$.

Answer: $2x \times \text{sin}(x) + (x^2 + 1) \times \text{cos}(x)$. Product Rule: $(2x) \times \sin(x) + (x^2 + 1) \times (\sin(x))'$.