The Product Rule - AP Calculus BC
Card 1 of 30
What is the derivative of $f(x) = x^3 \times \ln(x)$ using the Product Rule?
What is the derivative of $f(x) = x^3 \times \ln(x)$ using the Product Rule?
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$3x^2 \times \ln(x) + x^2$. Use $(fg)' = f'g + fg'$ where $\frac{d}{dx}[x^3] = 3x^2$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$.
$3x^2 \times \ln(x) + x^2$. Use $(fg)' = f'g + fg'$ where $\frac{d}{dx}[x^3] = 3x^2$ and $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$.
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Apply the Product Rule to find $f'(x)$ for $f(x) = (x + 1) \times \text{e}^x$.
Apply the Product Rule to find $f'(x)$ for $f(x) = (x + 1) \times \text{e}^x$.
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$e^x + (x + 1) \times e^x$. Factor out $e^x$ to get $e^x(1 + x + 1) = e^x(2 + x)$.
$e^x + (x + 1) \times e^x$. Factor out $e^x$ to get $e^x(1 + x + 1) = e^x(2 + x)$.
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Identify the derivative of $f(x) = e^x \times \text{sin}(x)$ using the Product Rule.
Identify the derivative of $f(x) = e^x \times \text{sin}(x)$ using the Product Rule.
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$e^x \times \text{sin}(x) + e^x \times \text{cos}(x)$. Use $(fg)' = f'g + fg'$ with derivatives of $e^x$ and $\sin(x)$.
$e^x \times \text{sin}(x) + e^x \times \text{cos}(x)$. Use $(fg)' = f'g + fg'$ with derivatives of $e^x$ and $\sin(x)$.
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Derive the function $f(x) = x^2 \times \text{e}^{2x}$ using the Product Rule.
Derive the function $f(x) = x^2 \times \text{e}^{2x}$ using the Product Rule.
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$2x \times e^{2x} + 2x^2 \times e^{2x}$. Factor out $2x e^{2x}$ to get $2x e^{2x}(1 + x)$.
$2x \times e^{2x} + 2x^2 \times e^{2x}$. Factor out $2x e^{2x}$ to get $2x e^{2x}(1 + x)$.
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What is the derivative of $f(x) = x^3 \times \text{sin}(x)$ using the Product Rule?
What is the derivative of $f(x) = x^3 \times \text{sin}(x)$ using the Product Rule?
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$3x^2 \times \text{sin}(x) + x^3 \times \text{cos}(x)$. Factor out $x^2$ to get $x^2(3\sin(x) + x\cos(x))$.
$3x^2 \times \text{sin}(x) + x^3 \times \text{cos}(x)$. Factor out $x^2$ to get $x^2(3\sin(x) + x\cos(x))$.
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Determine the derivative of $y = (3x) \times \text{e}^x$ using the Product Rule.
Determine the derivative of $y = (3x) \times \text{e}^x$ using the Product Rule.
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$3e^x + 3x \times e^x$. Factor out $3e^x$ to simplify to $3e^x(1 + x)$.
$3e^x + 3x \times e^x$. Factor out $3e^x$ to simplify to $3e^x(1 + x)$.
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Calculate the derivative using the Product Rule for $y = (1 - x) \times (2 + x)$.
Calculate the derivative using the Product Rule for $y = (1 - x) \times (2 + x)$.
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$-(2 + x) + (1 - x)$. Expand and simplify to get $-1 - 2x$.
$-(2 + x) + (1 - x)$. Expand and simplify to get $-1 - 2x$.
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What is the derivative of $f(x) = x \times \text{tan}(x)$ using the Product Rule?
What is the derivative of $f(x) = x \times \text{tan}(x)$ using the Product Rule?
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$\text{tan}(x) + x \times \text{sec}^2(x)$. Apply the product rule with derivatives $1$ and $\sec^2(x)$.
$\text{tan}(x) + x \times \text{sec}^2(x)$. Apply the product rule with derivatives $1$ and $\sec^2(x)$.
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Which option correctly applies the Product Rule to $f(x) = x^3 \times \text{cos}(x)$?
Which option correctly applies the Product Rule to $f(x) = x^3 \times \text{cos}(x)$?
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$3x^2 \times \text{cos}(x) - x^3 \times \text{sin}(x)$. Apply the product rule: derivative of first times second plus first times derivative of second.
$3x^2 \times \text{cos}(x) - x^3 \times \text{sin}(x)$. Apply the product rule: derivative of first times second plus first times derivative of second.
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Calculate the derivative using the Product Rule for $f(x) = 2x \times \text{sin}(x)$.
Calculate the derivative using the Product Rule for $f(x) = 2x \times \text{sin}(x)$.
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$2 \times \text{sin}(x) + 2x \times \text{cos}(x)$. Factor out $2$ to get $2(\sin(x) + x\cos(x))$.
$2 \times \text{sin}(x) + 2x \times \text{cos}(x)$. Factor out $2$ to get $2(\sin(x) + x\cos(x))$.
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Find the derivative using the Product Rule for $f(x) = x \times \text{cosh}(x)$.
Find the derivative using the Product Rule for $f(x) = x \times \text{cosh}(x)$.
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$\text{cosh}(x) + x \times \text{sinh}(x)$. Apply the product rule with hyperbolic function derivatives.
$\text{cosh}(x) + x \times \text{sinh}(x)$. Apply the product rule with hyperbolic function derivatives.
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Derive the function $f(x) = 5x \times \text{e}^{-x}$ using the Product Rule.
Derive the function $f(x) = 5x \times \text{e}^{-x}$ using the Product Rule.
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$5e^{-x} - 5x \times e^{-x}$. Factor out $5e^{-x}$ to get $5e^{-x}(1 - x)$.
$5e^{-x} - 5x \times e^{-x}$. Factor out $5e^{-x}$ to get $5e^{-x}(1 - x)$.
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Calculate the derivative of $f(x) = (x + 1) \times (x - 2)$ using the Product Rule.
Calculate the derivative of $f(x) = (x + 1) \times (x - 2)$ using the Product Rule.
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$(x - 2) + (x + 1)$. Simplifies to $2x - 1$ when expanded and combined.
$(x - 2) + (x + 1)$. Simplifies to $2x - 1$ when expanded and combined.
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Apply the Product Rule to $f(x) = \text{sin}(x) \times \text{cos}(x)$.
Apply the Product Rule to $f(x) = \text{sin}(x) \times \text{cos}(x)$.
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$\text{cos}^2(x) - \text{sin}^2(x)$. This simplifies to $\cos(2x)$ using the double angle identity.
$\text{cos}^2(x) - \text{sin}^2(x)$. This simplifies to $\cos(2x)$ using the double angle identity.
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Identify the derivative of $f(x) = x^3 \times \text{e}^x$ using the Product Rule.
Identify the derivative of $f(x) = x^3 \times \text{e}^x$ using the Product Rule.
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$3x^2 \times e^x + x^3 \times e^x$. Factor out $x^2 e^x$ to get $x^2 e^x(3 + x)$.
$3x^2 \times e^x + x^3 \times e^x$. Factor out $x^2 e^x$ to get $x^2 e^x(3 + x)$.
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What is the derivative of $f(x) = x \times \text{sin}(x)$ using the Product Rule?
What is the derivative of $f(x) = x \times \text{sin}(x)$ using the Product Rule?
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$\text{sin}(x) + x \times \text{cos}(x)$. Apply the product rule with derivatives $1$ and $\cos(x)$.
$\text{sin}(x) + x \times \text{cos}(x)$. Apply the product rule with derivatives $1$ and $\cos(x)$.
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Compute the derivative of $f(x) = x \times \text{cosh}(x)$ using the Product Rule.
Compute the derivative of $f(x) = x \times \text{cosh}(x)$ using the Product Rule.
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$\text{cosh}(x) + x \times \text{sinh}(x)$. Apply $(fg)' = f'g + fg'$ with hyperbolic function derivatives.
$\text{cosh}(x) + x \times \text{sinh}(x)$. Apply $(fg)' = f'g + fg'$ with hyperbolic function derivatives.
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Using the Product Rule, what is the derivative of $f(x) = x \times \text{log}(x)$?
Using the Product Rule, what is the derivative of $f(x) = x \times \text{log}(x)$?
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$\text{log}(x) + 1$. Use the fact that $\frac{d}{dx}[x \log(x)] = \log(x) + 1$.
$\text{log}(x) + 1$. Use the fact that $\frac{d}{dx}[x \log(x)] = \log(x) + 1$.
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Determine the derivative of $f(x) = x^2 \times \text{e}^x$ using the Product Rule.
Determine the derivative of $f(x) = x^2 \times \text{e}^x$ using the Product Rule.
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$2x \times e^x + x^2 \times e^x$. Factor out $x e^x$ to get $x e^x(2 + x)$.
$2x \times e^x + x^2 \times e^x$. Factor out $x e^x$ to get $x e^x(2 + x)$.
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Compute the derivative using the Product Rule for $y = x \times \text{cos}(x)$.
Compute the derivative using the Product Rule for $y = x \times \text{cos}(x)$.
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$\text{cos}(x) - x \times \text{sin}(x)$. Apply the product rule with derivatives $1$ and $-\sin(x)$.
$\text{cos}(x) - x \times \text{sin}(x)$. Apply the product rule with derivatives $1$ and $-\sin(x)$.
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Using the Product Rule, find the derivative for $y = x^2 \times \text{tan}(x)$.
Using the Product Rule, find the derivative for $y = x^2 \times \text{tan}(x)$.
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$2x \times \text{tan}(x) + x^2 \times \text{sec}^2(x)$. Factor out $x$ to get $x(2\tan(x) + x\sec^2(x))$.
$2x \times \text{tan}(x) + x^2 \times \text{sec}^2(x)$. Factor out $x$ to get $x(2\tan(x) + x\sec^2(x))$.
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What is the derivative of $f(x) = x^4 \times \text{tan}(x)$ using the Product Rule?
What is the derivative of $f(x) = x^4 \times \text{tan}(x)$ using the Product Rule?
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$4x^3 \times \text{tan}(x) + x^4 \times \text{sec}^2(x)$. Factor out $x^3$ to get $x^3(4\tan(x) + x\sec^2(x))$.
$4x^3 \times \text{tan}(x) + x^4 \times \text{sec}^2(x)$. Factor out $x^3$ to get $x^3(4\tan(x) + x\sec^2(x))$.
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Calculate the derivative using the Product Rule for $y = (2x) \times \text{ln}(x)$.
Calculate the derivative using the Product Rule for $y = (2x) \times \text{ln}(x)$.
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$2 \times \text{ln}(x) + \frac{2x}{x}$. Simplifies to $2\ln(x) + 2$ since $\frac{2x}{x} = 2$.
$2 \times \text{ln}(x) + \frac{2x}{x}$. Simplifies to $2\ln(x) + 2$ since $\frac{2x}{x} = 2$.
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Find the derivative of $f(x) = (x^2 + 1)(x^3 - x)$ using the Product Rule.
Find the derivative of $f(x) = (x^2 + 1)(x^3 - x)$ using the Product Rule.
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$(2x)(x^3 - x) + (x^2 + 1)(3x^2 - 1)$. Apply product rule to each factor and combine the terms.
$(2x)(x^3 - x) + (x^2 + 1)(3x^2 - 1)$. Apply product rule to each factor and combine the terms.
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What is the derivative of $f(x) = x^2 \times \text{ln}(x)$ using the Product Rule?
What is the derivative of $f(x) = x^2 \times \text{ln}(x)$ using the Product Rule?
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$2x \times \text{ln}(x) + x$. Apply $(fg)' = f'g + fg'$ with $f = x^2$ and $g = \ln(x)$.
$2x \times \text{ln}(x) + x$. Apply $(fg)' = f'g + fg'$ with $f = x^2$ and $g = \ln(x)$.
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State the formula for the Product Rule in calculus.
State the formula for the Product Rule in calculus.
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$(fg)' = f'g + fg'$. The fundamental formula where each function's derivative multiplies the other function.
$(fg)' = f'g + fg'$. The fundamental formula where each function's derivative multiplies the other function.
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What is the result of applying the Product Rule to $y = x \times e^{2x}$?
What is the result of applying the Product Rule to $y = x \times e^{2x}$?
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$e^{2x} + 2xe^{2x}$. Factor out $e^{2x}$ to get $e^{2x}(1 + 2x)$ after applying the rule.
$e^{2x} + 2xe^{2x}$. Factor out $e^{2x}$ to get $e^{2x}(1 + 2x)$ after applying the rule.
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Identify the derivative of $f(x) = x^5 \times \text{e}^{2x}$ using the Product Rule.
Identify the derivative of $f(x) = x^5 \times \text{e}^{2x}$ using the Product Rule.
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$5x^4 \times e^{2x} + 2x^5 \times e^{2x}$. Factor out $x^4 e^{2x}$ to get $x^4 e^{2x}(5 + 2x)$.
$5x^4 \times e^{2x} + 2x^5 \times e^{2x}$. Factor out $x^4 e^{2x}$ to get $x^4 e^{2x}(5 + 2x)$.
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Which is the correct application of the Product Rule to $f(x) = 3x \times \text{ln}(x)$?
Which is the correct application of the Product Rule to $f(x) = 3x \times \text{ln}(x)$?
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$3 \times \text{ln}(x) + \frac{3x}{x}$. Simplifies to $3 \ln(x) + 3$ since $\frac{3x}{x} = 3$.
$3 \times \text{ln}(x) + \frac{3x}{x}$. Simplifies to $3 \ln(x) + 3$ since $\frac{3x}{x} = 3$.
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What is the derivative of $f(x) = x^2 \times \text{e}^{-x}$ using the Product Rule?
What is the derivative of $f(x) = x^2 \times \text{e}^{-x}$ using the Product Rule?
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$2x \times e^{-x} - x^2 \times e^{-x}$. Factor out $x e^{-x}$ to get $x e^{-x}(2 - x)$.
$2x \times e^{-x} - x^2 \times e^{-x}$. Factor out $x e^{-x}$ to get $x e^{-x}(2 - x)$.
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