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

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

Flashcard 3: 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 4: Which rule differentiates the product of two differentiable functions?

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

Flashcard 5: Find $d/dx$ of $f(x) = x \times \text{cos}(x)$ using Product Rule.

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

Flashcard 6: 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 7: 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 8: Identify the derivative: $(3x^2) \times (2x^3)$. Use Product Rule.

Answer: $12x^4 + 18x^5$. Apply Product Rule: $f' = 6x, g' = 6x^2$, then $f'g + fg'$.

Flashcard 9: 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 10: 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 11: 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 12: Calculate the derivative of $h(x) = \text{ln}(x) \times \text{tan}(x)$.

Answer: $\frac{1}{x} \times \text{tan}(x) + \text{ln}(x) \times \text{sec}^2(x)$. Product Rule: $(\ln(x))' \times \tan(x) + \ln(x) \times (\tan(x))'$.

Flashcard 13: Determine the derivative: $y = (x^4 + x)(x^3 - 2)$.

Answer: $7x^6 - 2x^3 + 3x^4 - 2$. Product Rule: $(4x^3 + 1) \times (x^3 - 2) + (x^4 + x) \times 3x^2$.

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: 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 16: Identify the derivative: $(3x^2) \times (2x^3)$. Use Product Rule.

Answer: $12x^4 + 18x^5$. Apply Product Rule: $f' = 6x, g' = 6x^2$, then $f'g + fg'$.

Flashcard 17: 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 18: 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 19: What do $f(x)$ and $g(x)$ represent in the Product Rule?

Answer: Differentiable functions of $x$. They represent any two functions that can be differentiated.

Flashcard 20: 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 21: 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 22: 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 23: 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 24: Calculate the derivative of $y = (2x^2)(\text{cos}(x))$ using Product Rule.

Answer: $4x \text{cos}(x) - 2x^2 \text{sin}(x)$. Product Rule: $(2x^2)' \times \cos(x) + 2x^2 \times (\cos(x))'$.

Flashcard 25: 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 26: 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: $(2x) \times \sin(x) + (x^2 + 1) \times (\sin(x))'$.

Flashcard 27: 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 28: 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 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 $h(x) = x^2 \times \text{ln}(x)$.

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

Flashcard 31: 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 32: 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 33: 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 34: 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 35: Determine the derivative: $y = (x^4 + x)(x^3 - 2)$.

Answer: $7x^6 - 2x^3 + 3x^4 - 2$. Product Rule: $(4x^3 + 1) \times (x^3 - 2) + (x^4 + x) \times 3x^2$.

Flashcard 36: 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 37: 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 38: Which rule is used to differentiate the product of two functions?

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

Flashcard 39: Find $d/dx$ of $f(x) = x \times \text{cos}(x)$ using Product Rule.

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

Flashcard 40: 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 41: 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 42: 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 43: 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 44: Which rule is used to differentiate the product of two functions?

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

Flashcard 45: 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 46: 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 47: 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 48: Calculate the derivative of $h(x) = \text{ln}(x) \times \text{tan}(x)$.

Answer: $\frac{1}{x} \times \text{tan}(x) + \text{ln}(x) \times \text{sec}^2(x)$. Product Rule: $(\ln(x))' \times \tan(x) + \ln(x) \times (\tan(x))'$.

Flashcard 49: Which rule differentiates the product of two differentiable functions?

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

Flashcard 50: 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 51: What do $f(x)$ and $g(x)$ represent in the Product Rule?

Answer: Differentiable functions of $x$. They represent any two functions that can be differentiated.

Flashcard 52: 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 53: 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 54: 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 55: 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 56: 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 57: 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 58: Find the derivative of $y = (x^3 + 2x)(x^2 - 3)$ using Product Rule.

Answer: $5x^4 - 9x^2 + 4x - 6$. Product Rule: $(3x^2 + 2) \times (x^2 - 3) + (x^3 + 2x) \times 2x$.

Flashcard 59: 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 60: 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 61: 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: $(2x) \times \sin(x) + (x^2 + 1) \times (\sin(x))'$.

Flashcard 62: 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 63: 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 64: 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 65: 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 66: 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 67: 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 68: What is the derivative of $f(x) = x^2 \times \text{sin}(x)$?

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

Flashcard 69: 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 70: 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 71: Calculate the derivative of $y = (2x^2)(\text{cos}(x))$ using Product Rule.

Answer: $4x \text{cos}(x) - 2x^2 \text{sin}(x)$. Product Rule: $(2x^2)' \times \cos(x) + 2x^2 \times (\cos(x))'$.

Flashcard 72: 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 73: 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 74: 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 75: 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 76: Find the derivative of $y = (x^3 + 2x)(x^2 - 3)$ using Product Rule.

Answer: $5x^4 - 9x^2 + 4x - 6$. Product Rule: $(3x^2 + 2) \times (x^2 - 3) + (x^3 + 2x) \times 2x$.