The Quotient Rule - AP Calculus AB
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Which operation is used to combine $v\frac{du}{dx}$ and $u\frac{dv}{dx}$ in the Quotient Rule?
Which operation is used to combine $v\frac{du}{dx}$ and $u\frac{dv}{dx}$ in the Quotient Rule?
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Subtraction. The quotient rule formula uses subtraction between these two terms.
Subtraction. The quotient rule formula uses subtraction between these two terms.
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Use the Quotient Rule to differentiate $y = \frac{1}{x^2 + 4x}$.
Use the Quotient Rule to differentiate $y = \frac{1}{x^2 + 4x}$.
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$\frac{(x^2 + 4x)(0) - 1(2x + 4)}{(x^2 + 4x)^2}$. Since $u = 1$, its derivative is $0$, simplifying the numerator.
$\frac{(x^2 + 4x)(0) - 1(2x + 4)}{(x^2 + 4x)^2}$. Since $u = 1$, its derivative is $0$, simplifying the numerator.
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What is the denominator in the Quotient Rule for $\frac{u}{v}$?
What is the denominator in the Quotient Rule for $\frac{u}{v}$?
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$v^2$. The denominator function is always squared in the quotient rule.
$v^2$. The denominator function is always squared in the quotient rule.
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Which term is subtracted in the Quotient Rule formula?
Which term is subtracted in the Quotient Rule formula?
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$u\frac{dv}{dx}$. In the quotient rule formula, this term is subtracted from $v\frac{du}{dx}$.
$u\frac{dv}{dx}$. In the quotient rule formula, this term is subtracted from $v\frac{du}{dx}$.
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In the Quotient Rule, what happens to the denominator after differentiation?
In the Quotient Rule, what happens to the denominator after differentiation?
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It is squared. The original denominator $v$ becomes $v^2$ in the final result.
It is squared. The original denominator $v$ becomes $v^2$ in the final result.
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Evaluate the derivative: $y = \frac{2x^2}{x+3}$ using the Quotient Rule.
Evaluate the derivative: $y = \frac{2x^2}{x+3}$ using the Quotient Rule.
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$\frac{(x+3)(4x) - 2x^2(1)}{(x+3)^2}$. Apply quotient rule with $u = 2x^2$ and $v = x + 3$.
$\frac{(x+3)(4x) - 2x^2(1)}{(x+3)^2}$. Apply quotient rule with $u = 2x^2$ and $v = x + 3$.
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Differentiate $f(x) = \frac{x^2 + 1}{x}$ using the Quotient Rule.
Differentiate $f(x) = \frac{x^2 + 1}{x}$ using the Quotient Rule.
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$\frac{x(2x) - (x^2 + 1)(1)}{x^2}$. Apply quotient rule with $u = x^2 + 1$ and $v = x$.
$\frac{x(2x) - (x^2 + 1)(1)}{x^2}$. Apply quotient rule with $u = x^2 + 1$ and $v = x$.
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Differentiate $f(x) = \frac{x}{3x^2 + 1}$ using the Quotient Rule.
Differentiate $f(x) = \frac{x}{3x^2 + 1}$ using the Quotient Rule.
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$\frac{(3x^2+1)(1) - x(6x)}{(3x^2+1)^2}$. Apply quotient rule with $u = x$ and $v = 3x^2 + 1$.
$\frac{(3x^2+1)(1) - x(6x)}{(3x^2+1)^2}$. Apply quotient rule with $u = x$ and $v = 3x^2 + 1$.
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In the Quotient Rule, what does $\frac{d}{dx}$ denote?
In the Quotient Rule, what does $\frac{d}{dx}$ denote?
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Derivative with respect to $x$. This symbol indicates taking the derivative with respect to variable $x$.
Derivative with respect to $x$. This symbol indicates taking the derivative with respect to variable $x$.
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What operation is performed between $v\frac{du}{dx}$ and $u\frac{dv}{dx}$ in the Quotient Rule?
What operation is performed between $v\frac{du}{dx}$ and $u\frac{dv}{dx}$ in the Quotient Rule?
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Subtraction. The quotient rule requires subtracting $u\frac{dv}{dx}$ from $v\frac{du}{dx}$.
Subtraction. The quotient rule requires subtracting $u\frac{dv}{dx}$ from $v\frac{du}{dx}$.
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Identify the function that is squared in the denominator of the Quotient Rule.
Identify the function that is squared in the denominator of the Quotient Rule.
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$v$. The denominator function $v$ appears squared in the final quotient rule result.
$v$. The denominator function $v$ appears squared in the final quotient rule result.
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Evaluate the derivative: $y = \frac{x^2}{2x - 1}$ using the Quotient Rule.
Evaluate the derivative: $y = \frac{x^2}{2x - 1}$ using the Quotient Rule.
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$\frac{(2x-1)(2x) - x^2(2)}{(2x-1)^2}$. Apply quotient rule with $u = x^2$ and $v = 2x - 1$.
$\frac{(2x-1)(2x) - x^2(2)}{(2x-1)^2}$. Apply quotient rule with $u = x^2$ and $v = 2x - 1$.
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In the Quotient Rule, what is the role of $\frac{du}{dx}$?
In the Quotient Rule, what is the role of $\frac{du}{dx}$?
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Derivative of the numerator function. This term represents how the numerator changes with respect to $x$.
Derivative of the numerator function. This term represents how the numerator changes with respect to $x$.
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What is the derivative of $\frac{x^2}{x+1}$ using the Quotient Rule?
What is the derivative of $\frac{x^2}{x+1}$ using the Quotient Rule?
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$\frac{(x+1)(2x) - x^2(1)}{(x+1)^2}$. Apply quotient rule: $(x+1)$ times $2x$ minus $x^2$ times $1$, over $(x+1)^2$.
$\frac{(x+1)(2x) - x^2(1)}{(x+1)^2}$. Apply quotient rule: $(x+1)$ times $2x$ minus $x^2$ times $1$, over $(x+1)^2$.
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Evaluate the derivative: $y = \frac{5x^3}{2x + 1}$ using the Quotient Rule.
Evaluate the derivative: $y = \frac{5x^3}{2x + 1}$ using the Quotient Rule.
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$\frac{(2x+1)(15x^2) - 5x^3(2)}{(2x+1)^2}$. Apply quotient rule with $u = 5x^3$ and $v = 2x + 1$.
$\frac{(2x+1)(15x^2) - 5x^3(2)}{(2x+1)^2}$. Apply quotient rule with $u = 5x^3$ and $v = 2x + 1$.
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Differentiate $f(x) = \frac{2x}{x^2 + 3}$ using the Quotient Rule.
Differentiate $f(x) = \frac{2x}{x^2 + 3}$ using the Quotient Rule.
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$\frac{(x^2 + 3)(2) - 2x(2x)}{(x^2 + 3)^2}$. Apply quotient rule with $u = 2x$ and $v = x^2 + 3$.
$\frac{(x^2 + 3)(2) - 2x(2x)}{(x^2 + 3)^2}$. Apply quotient rule with $u = 2x$ and $v = x^2 + 3$.
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Evaluate the derivative: $y = \frac{4x^3}{x^2 - 2x}$ using the Quotient Rule.
Evaluate the derivative: $y = \frac{4x^3}{x^2 - 2x}$ using the Quotient Rule.
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$\frac{(x^2 - 2x)(12x^2) - 4x^3(2x - 2)}{(x^2 - 2x)^2}$. Apply quotient rule with $u = 4x^3$ and $v = x^2 - 2x$.
$\frac{(x^2 - 2x)(12x^2) - 4x^3(2x - 2)}{(x^2 - 2x)^2}$. Apply quotient rule with $u = 4x^3$ and $v = x^2 - 2x$.
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In the Quotient Rule, what does $u$ represent for $\frac{u}{v}$?
In the Quotient Rule, what does $u$ represent for $\frac{u}{v}$?
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Numerator function. This is the function in the numerator of the fraction $\frac{u}{v}$.
Numerator function. This is the function in the numerator of the fraction $\frac{u}{v}$.
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Differentiate $f(x) = \frac{x^2 - 1}{x^3}$ using the Quotient Rule.
Differentiate $f(x) = \frac{x^2 - 1}{x^3}$ using the Quotient Rule.
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$\frac{x^3(2x) - (x^2 - 1)(3x^2)}{x^6}$. Apply quotient rule with $u = x^2 - 1$ and $v = x^3$.
$\frac{x^3(2x) - (x^2 - 1)(3x^2)}{x^6}$. Apply quotient rule with $u = x^2 - 1$ and $v = x^3$.
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What is the Quotient Rule derivative for $\frac{1}{x^2}$?
What is the Quotient Rule derivative for $\frac{1}{x^2}$?
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$\frac{0 - 2x}{x^4}$. Since $u = 1$ has derivative $0$, only the subtraction term remains.
$\frac{0 - 2x}{x^4}$. Since $u = 1$ has derivative $0$, only the subtraction term remains.
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What is the result of differentiating $y = \frac{1}{x}$ using the Quotient Rule?
What is the result of differentiating $y = \frac{1}{x}$ using the Quotient Rule?
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$\frac{0 - 1}{x^2}$. Since $\frac{du}{dx} = 0$ for constant numerator, only the second term remains.
$\frac{0 - 1}{x^2}$. Since $\frac{du}{dx} = 0$ for constant numerator, only the second term remains.
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In the Quotient Rule, what does $v$ represent for $\frac{u}{v}$?
In the Quotient Rule, what does $v$ represent for $\frac{u}{v}$?
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Denominator function. This is the function in the denominator of the fraction $\frac{u}{v}$.
Denominator function. This is the function in the denominator of the fraction $\frac{u}{v}$.
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In the Quotient Rule, what is the derivative of the numerator?
In the Quotient Rule, what is the derivative of the numerator?
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$\frac{du}{dx}$. This represents taking the derivative of the numerator function $u$.
$\frac{du}{dx}$. This represents taking the derivative of the numerator function $u$.
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Differentiate $f(x) = \frac{3x^2 + 2}{x+4}$ using the Quotient Rule.
Differentiate $f(x) = \frac{3x^2 + 2}{x+4}$ using the Quotient Rule.
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$\frac{(x+4)(6x) - (3x^2+2)(1)}{(x+4)^2}$. Apply quotient rule with $u = 3x^2 + 2$ and $v = x + 4$.
$\frac{(x+4)(6x) - (3x^2+2)(1)}{(x+4)^2}$. Apply quotient rule with $u = 3x^2 + 2$ and $v = x + 4$.
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Differentiate $f(x) = \frac{x^2 + 2x}{5x + 1}$ using the Quotient Rule.
Differentiate $f(x) = \frac{x^2 + 2x}{5x + 1}$ using the Quotient Rule.
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$\frac{(5x+1)(2x+2) - (x^2+2x)(5)}{(5x+1)^2}$. Apply quotient rule with $u = x^2 + 2x$ and $v = 5x + 1$.
$\frac{(5x+1)(2x+2) - (x^2+2x)(5)}{(5x+1)^2}$. Apply quotient rule with $u = x^2 + 2x$ and $v = 5x + 1$.
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Evaluate the derivative: $y = \frac{7}{x^3 + 2x}$ using the Quotient Rule.
Evaluate the derivative: $y = \frac{7}{x^3 + 2x}$ using the Quotient Rule.
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$\frac{0(x^3 + 2x) - 7(3x^2 + 2)}{(x^3 + 2x)^2}$. Since $u = 7$ is constant, $\frac{du}{dx} = 0$ simplifies the expression.
$\frac{0(x^3 + 2x) - 7(3x^2 + 2)}{(x^3 + 2x)^2}$. Since $u = 7$ is constant, $\frac{du}{dx} = 0$ simplifies the expression.
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What is the derivative of $\frac{2x + 3}{x}$ using the Quotient Rule?
What is the derivative of $\frac{2x + 3}{x}$ using the Quotient Rule?
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$\frac{x(2) - (2x+3)(1)}{x^2}$. Apply quotient rule with $u = 2x + 3$ and $v = x$.
$\frac{x(2) - (2x+3)(1)}{x^2}$. Apply quotient rule with $u = 2x + 3$ and $v = x$.
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What is the derivative of $\frac{x}{x+2}$ using the Quotient Rule?
What is the derivative of $\frac{x}{x+2}$ using the Quotient Rule?
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$\frac{(x+2)(1) - x(1)}{(x+2)^2}$. Apply quotient rule with $u = x$ and $v = x + 2$.
$\frac{(x+2)(1) - x(1)}{(x+2)^2}$. Apply quotient rule with $u = x$ and $v = x + 2$.
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When using the Quotient Rule, what must be done to the denominator function $v$?
When using the Quotient Rule, what must be done to the denominator function $v$?
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Square it. The denominator $v$ must be squared to complete the quotient rule formula.
Square it. The denominator $v$ must be squared to complete the quotient rule formula.
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Differentiate $f(x) = \frac{x^3 + x}{x^2}$ using the Quotient Rule.
Differentiate $f(x) = \frac{x^3 + x}{x^2}$ using the Quotient Rule.
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$\frac{x^2(3x^2 + 1) - (x^3 + x)(2x)}{x^4}$. Apply quotient rule with $u = x^3 + x$ and $v = x^2$.
$\frac{x^2(3x^2 + 1) - (x^3 + x)(2x)}{x^4}$. Apply quotient rule with $u = x^3 + x$ and $v = x^2$.
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