The Product Rule - AP Calculus AB
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State the Product Rule formula for differentiation.
State the Product Rule formula for differentiation.
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$ (f \times g)' = f' \times g + f \times g' $. Standard formula where each function multiplies the other's derivative.
$ (f \times g)' = f' \times g + f \times g' $. Standard formula where each function multiplies the other's derivative.
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What is the outcome of differentiating $f(x) = xe^x$ using Product Rule?
What is the outcome of differentiating $f(x) = xe^x$ using Product Rule?
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$e^x + xe^x$. Product Rule: $1 \times e^x + x \times e^x$.
$e^x + xe^x$. Product Rule: $1 \times e^x + x \times e^x$.
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Find $d/dx$ of $f(x) = x^2 \times \text{sec}(x)$. Use Product Rule.
Find $d/dx$ of $f(x) = x^2 \times \text{sec}(x)$. Use Product Rule.
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$2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.
$2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.
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Does the Product Rule apply to $f(x) = x^2 \times x^3$?
Does the Product Rule apply to $f(x) = x^2 \times x^3$?
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Yes, but it can be simplified before applying. Could simplify to $x^5$ first, but Product Rule still applies.
Yes, but it can be simplified before applying. Could simplify to $x^5$ first, but Product Rule still applies.
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Which rule differentiates the product of two differentiable functions?
Which rule differentiates the product of two differentiable functions?
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The Product Rule. Standard rule for finding derivatives of function products.
The Product Rule. Standard rule for finding derivatives of function products.
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What is $d/dx$ for $h(x) = x^3 \times \text{ln}(x)$ using the Product Rule?
What is $d/dx$ for $h(x) = x^3 \times \text{ln}(x)$ using the Product Rule?
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$3x^2 \text{ln}(x) + x^2$. Product Rule: $(x^3)' \times \ln(x) + x^3 \times (\ln(x))'$.
$3x^2 \text{ln}(x) + x^2$. Product Rule: $(x^3)' \times \ln(x) + x^3 \times (\ln(x))'$.
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Calculate the derivative of $h(x) = x \text{ln}(x)$.
Calculate the derivative of $h(x) = x \text{ln}(x)$.
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$1 \times \text{ln}(x) + \frac{x}{x}$. Product Rule: $1 \times \ln(x) + x \times \frac{1}{x}$.
$1 \times \text{ln}(x) + \frac{x}{x}$. Product Rule: $1 \times \ln(x) + x \times \frac{1}{x}$.
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What is the result of differentiating $u(x) = e^x \times \text{ln}(x)$?
What is the result of differentiating $u(x) = e^x \times \text{ln}(x)$?
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$e^x \times \text{ln}(x) + \frac{e^x}{x}$. Product Rule: $(e^x)' \times \ln(x) + e^x \times (\ln(x))'$.
$e^x \times \text{ln}(x) + \frac{e^x}{x}$. Product Rule: $(e^x)' \times \ln(x) + e^x \times (\ln(x))'$.
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Identify the derivative of $y = (3x + 2)(x^3 - 1)$ using the Product Rule.
Identify the derivative of $y = (3x + 2)(x^3 - 1)$ using the Product Rule.
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$9x^3 + 6x^2 - 3x - 2$. Product Rule: $3 \times (x^3 - 1) + (3x + 2) \times 3x^2$.
$9x^3 + 6x^2 - 3x - 2$. Product Rule: $3 \times (x^3 - 1) + (3x + 2) \times 3x^2$.
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What is the derivative of $h(x) = x^2 \times \text{sec}(x)$ using Product Rule?
What is the derivative of $h(x) = x^2 \times \text{sec}(x)$ using Product Rule?
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$2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.
$2x \times \text{sec}(x) + x^2 \times \text{sec}(x)\text{tan}(x)$. Product Rule: $(x^2)' \times \sec(x) + x^2 \times (\sec(x))'$.
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Evaluate the derivative: $h(x) = x^2 e^x$ using the Product Rule.
Evaluate the derivative: $h(x) = x^2 e^x$ using the Product Rule.
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$2x e^x + x^2 e^x$. Product Rule: $(x^2)' \times e^x + x^2 \times (e^x)'$.
$2x e^x + x^2 e^x$. Product Rule: $(x^2)' \times e^x + x^2 \times (e^x)'$.
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Is the Product Rule applicable to $f(x) = (x^2 + x)(x - 1)$?
Is the Product Rule applicable to $f(x) = (x^2 + x)(x - 1)$?
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Yes, it is applicable. Two differentiable functions multiplied together require Product Rule.
Yes, it is applicable. Two differentiable functions multiplied together require Product Rule.
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Calculate the derivative for $y = (\text{sin}(x))(x^2)$ using Product Rule.
Calculate the derivative for $y = (\text{sin}(x))(x^2)$ using Product Rule.
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$2x \times \text{sin}(x) + x^2 \times \text{cos}(x)$. Product Rule applied with order switched: same result.
$2x \times \text{sin}(x) + x^2 \times \text{cos}(x)$. Product Rule applied with order switched: same result.
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What is the derivative of $f(x) = (x + 2)(x^2 + 1)$?
What is the derivative of $f(x) = (x + 2)(x^2 + 1)$?
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$3x^2 + 4x + 2$. Product Rule: $1 \times (x^2 + 1) + (x + 2) \times 2x$.
$3x^2 + 4x + 2$. Product Rule: $1 \times (x^2 + 1) + (x + 2) \times 2x$.
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Find the derivative: $f(x) = (2x + 1)(x^2 - 4x + 4)$. Use Product Rule.
Find the derivative: $f(x) = (2x + 1)(x^2 - 4x + 4)$. Use Product Rule.
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$2x^3 - 6x^2 + 10x - 4$. Product Rule: $2 \times (x^2 - 4x + 4) + (2x + 1) \times (2x - 4)$.
$2x^3 - 6x^2 + 10x - 4$. Product Rule: $2 \times (x^2 - 4x + 4) + (2x + 1) \times (2x - 4)$.
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Find the derivative of $y = (x + 3)(x^2 - x + 1)$.
Find the derivative of $y = (x + 3)(x^2 - x + 1)$.
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$3x^2 - 2x + 2$. Product Rule: $1 \times (x^2 - x + 1) + (x + 3) \times (2x - 1)$.
$3x^2 - 2x + 2$. Product Rule: $1 \times (x^2 - x + 1) + (x + 3) \times (2x - 1)$.
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What is $d/dx$ of $f(x) = (x^2)(\text{cos}(x))$ using Product Rule?
What is $d/dx$ of $f(x) = (x^2)(\text{cos}(x))$ using Product Rule?
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$2x \text{cos}(x) - x^2 \text{sin}(x)$. Product Rule: $(x^2)' \times \cos(x) + x^2 \times (\cos(x))'$.
$2x \text{cos}(x) - x^2 \text{sin}(x)$. Product Rule: $(x^2)' \times \cos(x) + x^2 \times (\cos(x))'$.
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Calculate the derivative: $y = (x^4)(\text{sin}(x))$ using Product Rule.
Calculate the derivative: $y = (x^4)(\text{sin}(x))$ using Product Rule.
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$4x^3 \text{sin}(x) + x^4 \text{cos}(x)$. Product Rule: $(x^4)' \times \sin(x) + x^4 \times (\sin(x))'$.
$4x^3 \text{sin}(x) + x^4 \text{cos}(x)$. Product Rule: $(x^4)' \times \sin(x) + x^4 \times (\sin(x))'$.
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Find the derivative of $h(x) = x^2 \times \text{ln}(x)$.
Find the derivative of $h(x) = x^2 \times \text{ln}(x)$.
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$2x \text{ln}(x) + x$. Product Rule: $(x^2)' \times \ln(x) + x^2 \times (\ln(x))'$.
$2x \text{ln}(x) + x$. Product Rule: $(x^2)' \times \ln(x) + x^2 \times (\ln(x))'$.
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Is the Product Rule used for $f(x) = (x + 1)(x^2 - 1)$?
Is the Product Rule used for $f(x) = (x + 1)(x^2 - 1)$?
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Yes, it is applicable. Product of two functions requires the Product Rule.
Yes, it is applicable. Product of two functions requires the Product Rule.
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Evaluate the derivative: $y = (x^3 + 2)(x^2 - 1)$ using Product Rule.
Evaluate the derivative: $y = (x^3 + 2)(x^2 - 1)$ using Product Rule.
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$5x^4 - 3x^2 + 4x - 2$. Product Rule: $(3x^2) \times (x^2 - 1) + (x^3 + 2) \times 2x$.
$5x^4 - 3x^2 + 4x - 2$. Product Rule: $(3x^2) \times (x^2 - 1) + (x^3 + 2) \times 2x$.
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Find the derivative of $y = (x^2 + 3x + 1)(x - 2)$.
Find the derivative of $y = (x^2 + 3x + 1)(x - 2)$.
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$3x^2 + 2x - 5$. Product Rule: $(2x + 3) \times (x - 2) + (x^2 + 3x + 1) \times 1$.
$3x^2 + 2x - 5$. Product Rule: $(2x + 3) \times (x - 2) + (x^2 + 3x + 1) \times 1$.
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Determine the derivative: $f(x) = (x^3)(\text{tan}(x))$. Use Product Rule.
Determine the derivative: $f(x) = (x^3)(\text{tan}(x))$. Use Product Rule.
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$3x^2 \text{tan}(x) + x^3 \text{sec}^2(x)$. Product Rule: $(x^3)' \times \tan(x) + x^3 \times (\tan(x))'$.
$3x^2 \text{tan}(x) + x^3 \text{sec}^2(x)$. Product Rule: $(x^3)' \times \tan(x) + x^3 \times (\tan(x))'$.
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Find $d/dx$ of $y = x(x^2 + 1)$ using Product Rule.
Find $d/dx$ of $y = x(x^2 + 1)$ using Product Rule.
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$3x^2 + 1$. Product Rule: $1 \times (x^2 + 1) + x \times 2x$.
$3x^2 + 1$. Product Rule: $1 \times (x^2 + 1) + x \times 2x$.
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Determine the derivative: $y = (x^5)(\text{cos}(x))$. Use Product Rule.
Determine the derivative: $y = (x^5)(\text{cos}(x))$. Use Product Rule.
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$5x^4 \text{cos}(x) - x^5 \text{sin}(x)$. Product Rule: $(x^5)' \times \cos(x) + x^5 \times (\cos(x))'$.
$5x^4 \text{cos}(x) - x^5 \text{sin}(x)$. Product Rule: $(x^5)' \times \cos(x) + x^5 \times (\cos(x))'$.
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What is the derivative of $y = (x^4)(e^x)$ using Product Rule?
What is the derivative of $y = (x^4)(e^x)$ using Product Rule?
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$4x^3 e^x + x^4 e^x$. Product Rule: $(x^4)' \times e^x + x^4 \times (e^x)'$.
$4x^3 e^x + x^4 e^x$. Product Rule: $(x^4)' \times e^x + x^4 \times (e^x)'$.
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Which rule is used to differentiate the product of two functions?
Which rule is used to differentiate the product of two functions?
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The Product Rule. Essential rule for differentiating products of functions.
The Product Rule. Essential rule for differentiating products of functions.
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Find the derivative: $y = (3x^2 + x)(x^2 - x)$ using Product Rule.
Find the derivative: $y = (3x^2 + x)(x^2 - x)$ using Product Rule.
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$9x^3 - 4x^2 + x$. Product Rule: $(6x + 1) \times (x^2 - x) + (3x^2 + x) \times (2x - 1)$.
$9x^3 - 4x^2 + x$. Product Rule: $(6x + 1) \times (x^2 - x) + (3x^2 + x) \times (2x - 1)$.
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Which component is found first in the Product Rule: $f'(x)$ or $g'(x)$?
Which component is found first in the Product Rule: $f'(x)$ or $g'(x)$?
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Either order is acceptable. The Product Rule is symmetric; order doesn't matter.
Either order is acceptable. The Product Rule is symmetric; order doesn't matter.
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Find the derivative of $f(x) = (x^2 + 1)(\text{sin}(x))$.
Find the derivative of $f(x) = (x^2 + 1)(\text{sin}(x))$.
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$2x \times \text{sin}(x) + (x^2 + 1) \times \text{cos}(x)$. Product Rule: $(2x) \times \sin(x) + (x^2 + 1) \times (\sin(x))'$.
$2x \times \text{sin}(x) + (x^2 + 1) \times \text{cos}(x)$. Product Rule: $(2x) \times \sin(x) + (x^2 + 1) \times (\sin(x))'$.
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