AP Calculus BC 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 BC.

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 the derivative of $y = x^2 \times \text{cos}(x)$ using the Product Rule.

Answer: $2x \times \text{cos}(x) - x^2 \times \text{sin}(x)$. Factor out $x$ to get $x(2\cos(x) - x\sin(x))$.

Flashcard 2: Determine the derivative of $f(x) = x^2 \times \text{e}^x$ using the Product Rule.

Answer: $2x \times e^x + x^2 \times e^x$. Factor out $x e^x$ to get $x e^x(2 + x)$.

Flashcard 3: What is the derivative of $f(x) = x^2 \times \text{ln}(x)$ using the Product Rule?

Answer: $2x \times \text{ln}(x) + x$. Apply $(fg)' = f'g + fg'$ with $f = x^2$ and $g = \ln(x)$.

Flashcard 4: Identify the derivative of $f(x) = e^x \times \text{sin}(x)$ using the Product Rule.

Answer: $e^x \times \text{sin}(x) + e^x \times \text{cos}(x)$. Use $(fg)' = f'g + fg'$ with derivatives of $e^x$ and $\sin(x)$.

Flashcard 5: What is the derivative of $f(x) = x^2 \times \text{e}^{-x}$ using the Product Rule?

Answer: $2x \times e^{-x} - x^2 \times e^{-x}$. Factor out $x e^{-x}$ to get $x e^{-x}(2 - x)$.

Flashcard 6: What is the derivative of $f(x) = x^4 \times \text{tan}(x)$ using the Product Rule?

Answer: $4x^3 \times \text{tan}(x) + x^4 \times \text{sec}^2(x)$. Factor out $x^3$ to get $x^3(4\tan(x) + x\sec^2(x))$.

Flashcard 7: What is the derivative of $f(x) = x^2 \times \text{e}^{3x}$ using the Product Rule?

Answer: $2x \times \text{e}^{3x} + 3x^2 \times \text{e}^{3x}$. Factor out $x e^{3x}$ to get $x e^{3x}(2 + 3x)$.

Flashcard 8: What is the derivative of $f(x) = x^4 \times \text{tan}(x)$ using the Product Rule?

Answer: $4x^3 \times \text{tan}(x) + x^4 \times \text{sec}^2(x)$. Factor out $x^3$ to get $x^3(4\tan(x) + x\sec^2(x))$.

Flashcard 9: Compute the derivative of $f(x) = x \times \text{cosh}(x)$ using the Product Rule.

Answer: $\text{cosh}(x) + x \times \text{sinh}(x)$. Apply $(fg)' = f'g + fg'$ with hyperbolic function derivatives.

Flashcard 10: Apply the Product Rule to find $f'(x)$ for $f(x) = x \times \text{e}^x$.

Answer: $e^x + x \times e^x$. Factor out $e^x$ to get $e^x(1 + x)$.

Flashcard 11: Identify the derivative of $f(x) = x^3 \times \text{e}^x$ using the Product Rule.

Answer: $3x^2 \times e^x + x^3 \times e^x$. Factor out $x^2 e^x$ to get $x^2 e^x(3 + x)$.

Flashcard 12: Which option correctly applies the Product Rule to $f(x) = x^3 \times \text{cos}(x)$?

Answer: $3x^2 \times \text{cos}(x) - x^3 \times \text{sin}(x)$. Apply the product rule: derivative of first times second plus first times derivative of second.

Flashcard 13: Identify the derivative of $f(x) = x^5 \times \text{e}^{2x}$ using the Product Rule.

Answer: $5x^4 \times e^{2x} + 2x^5 \times e^{2x}$. Factor out $x^4 e^{2x}$ to get $x^4 e^{2x}(5 + 2x)$.

Flashcard 14: Compute the derivative of $f(x) = x \times \text{cosh}(x)$ using the Product Rule.

Answer: $\text{cosh}(x) + x \times \text{sinh}(x)$. Apply $(fg)' = f'g + fg'$ with hyperbolic function derivatives.

Flashcard 15: Derive the function $f(x) = x^2 \times \text{e}^{2x}$ using the Product Rule.

Answer: $2x \times e^{2x} + 2x^2 \times e^{2x}$. Factor out $2x e^{2x}$ to get $2x e^{2x}(1 + x)$.

Flashcard 16: Identify the derivative of $f(x) = x^5 \times \text{e}^{2x}$ using the Product Rule.

Answer: $5x^4 \times e^{2x} + 2x^5 \times e^{2x}$. Factor out $x^4 e^{2x}$ to get $x^4 e^{2x}(5 + 2x)$.

Flashcard 17: Using the Product Rule, what is the derivative of $f(x) = x \times \text{log}(x)$?

Answer: $\text{log}(x) + 1$. Use the fact that $\frac{d}{dx}[x \log(x)] = \log(x) + 1$.

Flashcard 18: Compute the derivative using the Product Rule for $y = x \times \text{cos}(x)$.

Answer: $\text{cos}(x) - x \times \text{sin}(x)$. Apply the product rule with derivatives $1$ and $-\sin(x)$.

Flashcard 19: Find the derivative of $f(x) = (x^2 + 1)(x^3 - x)$ using the Product Rule.

Answer: $(2x)(x^3 - x) + (x^2 + 1)(3x^2 - 1)$. Apply product rule to each factor and combine the terms.

Flashcard 20: Using the Product Rule, find the derivative for $y = x^2 \times \text{tan}(x)$.

Answer: $2x \times \text{tan}(x) + x^2 \times \text{sec}^2(x)$. Factor out $x$ to get $x(2\tan(x) + x\sec^2(x))$.

Flashcard 21: Find the derivative using the Product Rule for $f(x) = x \times \text{cosh}(x)$.

Answer: $\text{cosh}(x) + x \times \text{sinh}(x)$. Apply the product rule with hyperbolic function derivatives.

Flashcard 22: Calculate the derivative using the Product Rule for $f(x) = 2x \times \text{sin}(x)$.

Answer: $2 \times \text{sin}(x) + 2x \times \text{cos}(x)$. Factor out $2$ to get $2(\sin(x) + x\cos(x))$.

Flashcard 23: Identify the derivative of $f(x) = e^x \times \text{sin}(x)$ using the Product Rule.

Answer: $e^x \times \text{sin}(x) + e^x \times \text{cos}(x)$. Use $(fg)' = f'g + fg'$ with derivatives of $e^x$ and $\sin(x)$.

Flashcard 24: Derive the function $f(x) = 5x \times \text{e}^{-x}$ using the Product Rule.

Answer: $5e^{-x} - 5x \times e^{-x}$. Factor out $5e^{-x}$ to get $5e^{-x}(1 - x)$.

Flashcard 25: What is the derivative of $f(x) = x^2 \times \text{ln}(x)$ using the Product Rule?

Answer: $2x \times \text{ln}(x) + x$. Apply $(fg)' = f'g + fg'$ with $f = x^2$ and $g = \ln(x)$.

Flashcard 26: Derive the function $f(x) = 5x \times \text{e}^{-x}$ using the Product Rule.

Answer: $5e^{-x} - 5x \times e^{-x}$. Factor out $5e^{-x}$ to get $5e^{-x}(1 - x)$.

Flashcard 27: Find the derivative of $f(x) = (x^2 + 1)(x^3 - x)$ using the Product Rule.

Answer: $(2x)(x^3 - x) + (x^2 + 1)(3x^2 - 1)$. Apply product rule to each factor and combine the terms.

Flashcard 28: Calculate the derivative using the Product Rule for $y = (1 - x) \times (2 + x)$.

Answer: $-(2 + x) + (1 - x)$. Expand and simplify to get $-1 - 2x$.

Flashcard 29: Determine the derivative of $y = (3x) \times \text{e}^x$ using the Product Rule.

Answer: $3e^x + 3x \times e^x$. Factor out $3e^x$ to simplify to $3e^x(1 + x)$.

Flashcard 30: Apply the Product Rule to find $f'(x)$ for $f(x) = (x + 1) \times \text{e}^x$.

Answer: $e^x + (x + 1) \times e^x$. Factor out $e^x$ to get $e^x(1 + x + 1) = e^x(2 + x)$.

Flashcard 31: What is the derivative of $f(x) = x \times \text{sin}(x)$ using the Product Rule?

Answer: $\text{sin}(x) + x \times \text{cos}(x)$. Apply the product rule with derivatives $1$ and $\cos(x)$.

Flashcard 32: What is the derivative of $f(x) = x^3 \times \text{sin}(x)$ using the Product Rule?

Answer: $3x^2 \times \text{sin}(x) + x^3 \times \text{cos}(x)$. Factor out $x^2$ to get $x^2(3\sin(x) + x\cos(x))$.

Flashcard 33: What is the derivative of $f(x) = x \times \text{tan}(x)$ using the Product Rule?

Answer: $\text{tan}(x) + x \times \text{sec}^2(x)$. Apply the product rule with derivatives $1$ and $\sec^2(x)$.

Flashcard 34: What is the derivative of $f(x) = x^3 \times \text{sin}(x)$ using the Product Rule?

Answer: $3x^2 \times \text{sin}(x) + x^3 \times \text{cos}(x)$. Factor out $x^2$ to get $x^2(3\sin(x) + x\cos(x))$.

Flashcard 35: Derive the function $f(x) = x^2 \times \text{e}^{2x}$ using the Product Rule.

Answer: $2x \times e^{2x} + 2x^2 \times e^{2x}$. Factor out $2x e^{2x}$ to get $2x e^{2x}(1 + x)$.

Flashcard 36: Calculate the derivative using the Product Rule for $y = (2x) \times \text{ln}(x)$.

Answer: $2 \times \text{ln}(x) + \frac{2x}{x}$. Simplifies to $2\ln(x) + 2$ since $\frac{2x}{x} = 2$.

Flashcard 37: What is the derivative of $f(x) = x \times \text{sin}(x)$ using the Product Rule?

Answer: $\text{sin}(x) + x \times \text{cos}(x)$. Apply the product rule with derivatives $1$ and $\cos(x)$.

Flashcard 38: Find the derivative using the Product Rule for $f(x) = x \times \text{cosh}(x)$.

Answer: $\text{cosh}(x) + x \times \text{sinh}(x)$. Apply the product rule with hyperbolic function derivatives.

Flashcard 39: What is the derivative of $f(x) = x^2 \times \text{e}^{3x}$ using the Product Rule?

Answer: $2x \times \text{e}^{3x} + 3x^2 \times \text{e}^{3x}$. Factor out $x e^{3x}$ to get $x e^{3x}(2 + 3x)$.

Flashcard 40: Apply the Product Rule to $f(x) = \text{sin}(x) \times \text{cos}(x)$.

Answer: $\text{cos}^2(x) - \text{sin}^2(x)$. This simplifies to $\cos(2x)$ using the double angle identity.

Flashcard 41: Find the derivative of $y = x^2 \times \text{cos}(x)$ using the Product Rule.

Answer: $2x \times \text{cos}(x) - x^2 \times \text{sin}(x)$. Factor out $x$ to get $x(2\cos(x) - x\sin(x))$.

Flashcard 42: What is the derivative of $f(x) = x^2 \times \text{e}^{-x}$ using the Product Rule?

Answer: $2x \times e^{-x} - x^2 \times e^{-x}$. Factor out $x e^{-x}$ to get $x e^{-x}(2 - x)$.

Flashcard 43: Apply the Product Rule to $f(x) = \text{sin}(x) \times \text{cos}(x)$.

Answer: $\text{cos}^2(x) - \text{sin}^2(x)$. This simplifies to $\cos(2x)$ using the double angle identity.

Flashcard 44: What is the result of applying the Product Rule to $y = x \times e^{2x}$?

Answer: $e^{2x} + 2xe^{2x}$. Factor out $e^{2x}$ to get $e^{2x}(1 + 2x)$ after applying the rule.

Flashcard 45: Apply the Product Rule to find $f'(x)$ for $f(x) = x \times \text{e}^x$.

Answer: $e^x + x \times e^x$. Factor out $e^x$ to get $e^x(1 + x)$.

Flashcard 46: What is the derivative of $f(x) = x^3 \times \ln(x)$ using the Product Rule?

Answer: $3x^2 \times \ln(x) + x^2$. Use $(fg)' = f'g + fg'$ where $\frac{d}{dx}[x^3] = 3x^2$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$.

Flashcard 47: Identify the derivative of $f(x) = x^3 \times \text{e}^x$ using the Product Rule.

Answer: $3x^2 \times e^x + x^3 \times e^x$. Factor out $x^2 e^x$ to get $x^2 e^x(3 + x)$.

Flashcard 48: State the formula for the Product Rule in calculus.

Answer: $(fg)' = f'g + fg'$. The fundamental formula where each function's derivative multiplies the other function.

Flashcard 49: What is the derivative of $f(x) = x \times \text{tan}(x)$ using the Product Rule?

Answer: $\text{tan}(x) + x \times \text{sec}^2(x)$. Apply the product rule with derivatives $1$ and $\sec^2(x)$.

Flashcard 50: Calculate the derivative of $f(x) = (x + 1) \times (x - 2)$ using the Product Rule.

Answer: $(x - 2) + (x + 1)$. Simplifies to $2x - 1$ when expanded and combined.

Flashcard 51: Using the Product Rule, find the derivative for $y = x^2 \times \text{tan}(x)$.

Answer: $2x \times \text{tan}(x) + x^2 \times \text{sec}^2(x)$. Factor out $x$ to get $x(2\tan(x) + x\sec^2(x))$.

Flashcard 52: State the formula for the Product Rule in calculus.

Answer: $(fg)' = f'g + fg'$. The fundamental formula where each function's derivative multiplies the other function.

Flashcard 53: Calculate the derivative of $f(x) = (x + 1) \times (x - 2)$ using the Product Rule.

Answer: $(x - 2) + (x + 1)$. Simplifies to $2x - 1$ when expanded and combined.

Flashcard 54: Which is the correct application of the Product Rule to $f(x) = 3x \times \text{ln}(x)$?

Answer: $3 \times \text{ln}(x) + \frac{3x}{x}$. Simplifies to $3\ln(x) + 3$ since $\frac{3x}{x} = 3$.

Flashcard 55: What is the derivative of $f(x) = x^3 \times \text{ln}(x)$ using the Product Rule?

Answer: $3x^2 \times \text{ln}(x) + x^2$. Use $(fg)' = f'g + fg'$ where $\frac{d}{dx}[x^3] = 3x^2$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$.

Flashcard 56: Calculate the derivative using the Product Rule for $y = (2x) \times \text{ln}(x)$.

Answer: $2 \times \text{ln}(x) + \frac{2x}{x}$. Simplifies to $2\ln(x) + 2$ since $\frac{2x}{x} = 2$.

Flashcard 57: Calculate the derivative using the Product Rule for $y = (1 - x) \times (2 + x)$.

Answer: $-(2 + x) + (1 - x)$. Expand and simplify to get $-1 - 2x$.

Flashcard 58: Which is the correct application of the Product Rule to $f(x) = 3x \times \text{ln}(x)$?

Answer: $3 \times \text{ln}(x) + \frac{3x}{x}$. Simplifies to $3 \ln(x) + 3$ since $\frac{3x}{x} = 3$.

Flashcard 59: What is the result of applying the Product Rule to $y = x \times e^{2x}$?

Answer: $e^{2x} + 2xe^{2x}$. Factor out $e^{2x}$ to get $e^{2x}(1 + 2x)$ after applying the rule.

Flashcard 60: Compute the derivative using the Product Rule for $y = x \times \text{cos}(x)$.

Answer: $\text{cos}(x) - x \times \text{sin}(x)$. Apply the product rule with derivatives $1$ and $-\sin(x)$.

Flashcard 61: Using the Product Rule, what is the derivative of $f(x) = x \times \text{log}(x)$?

Answer: $\text{log}(x) + 1$. Use the fact that $\frac{d}{dx}[x \log(x)] = \log(x) + 1$.

Flashcard 62: Apply the Product Rule to find $f'(x)$ for $f(x) = (x + 1) \times \text{e}^x$.

Answer: $e^x + (x + 1) \times e^x$. Factor out $e^x$ to get $e^x(1 + x + 1) = e^x(2 + x)$.

Flashcard 63: Determine the derivative of $f(x) = x^2 \times \text{e}^x$ using the Product Rule.

Answer: $2x \times e^x + x^2 \times e^x$. Factor out $x e^x$ to get $x e^x(2 + x)$.

Flashcard 64: Determine the derivative of $y = (3x) \times \text{e}^x$ using the Product Rule.

Answer: $3e^x + 3x \times e^x$. Factor out $3e^x$ to simplify to $3e^x(1 + x)$.

Flashcard 65: Calculate the derivative using the Product Rule for $f(x) = 2x \times \text{sin}(x)$.

Answer: $2 \times \text{sin}(x) + 2x \times \text{cos}(x)$. Factor out $2$ to get $2(\sin(x) + x\cos(x))$.