The Quotient Rule - AP Calculus BC
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What is the derivative of $\frac{x}{e^x}$ using the Quotient Rule?
What is the derivative of $\frac{x}{e^x}$ using the Quotient Rule?
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$\frac{e^x \cdot 1 - x \cdot e^x}{(e^x)^2}$. Quotient rule with exponential denominator.
$\frac{e^x \cdot 1 - x \cdot e^x}{(e^x)^2}$. Quotient rule with exponential denominator.
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Find the derivative of $\frac{2x+3}{x^2}$ using the Quotient Rule.
Find the derivative of $\frac{2x+3}{x^2}$ using the Quotient Rule.
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$\frac{x^2 \cdot 2 - (2x+3) \cdot 2x}{x^4}$. Quotient rule setup for rational function.
$\frac{x^2 \cdot 2 - (2x+3) \cdot 2x}{x^4}$. Quotient rule setup for rational function.
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What is $v'$ if $v = \sin x$ in the Quotient Rule?
What is $v'$ if $v = \sin x$ in the Quotient Rule?
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$v' = \cos x$. Derivative of sine function.
$v' = \cos x$. Derivative of sine function.
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What is the derivative of $\frac{\ln x}{x}$ using the Quotient Rule?
What is the derivative of $\frac{\ln x}{x}$ using the Quotient Rule?
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$\frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2}$. Quotient rule with logarithmic numerator.
$\frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2}$. Quotient rule with logarithmic numerator.
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Find $u'$ for $u = \cos x$ in the Quotient Rule.
Find $u'$ for $u = \cos x$ in the Quotient Rule.
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$u' = -\sin x$. Derivative of cosine is negative sine.
$u' = -\sin x$. Derivative of cosine is negative sine.
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What is $v'$ if $v = x^3$ in the Quotient Rule?
What is $v'$ if $v = x^3$ in the Quotient Rule?
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$v' = 3x^2$. Power rule applied to cubic function.
$v' = 3x^2$. Power rule applied to cubic function.
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What is $u'$ if $u = e^x$ in the Quotient Rule?
What is $u'$ if $u = e^x$ in the Quotient Rule?
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$u' = e^x$. Exponential function derivative.
$u' = e^x$. Exponential function derivative.
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Identify $u$ and $v$ for $\frac{e^x}{\sin x}$ in the Quotient Rule.
Identify $u$ and $v$ for $\frac{e^x}{\sin x}$ in the Quotient Rule.
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$u = e^x$, $v = \sin x$. Numerator and denominator identification for quotient rule.
$u = e^x$, $v = \sin x$. Numerator and denominator identification for quotient rule.
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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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$\frac{(x+1)\cdot 2x - x^2 \cdot 1}{(x+1)^2}$. Applying quotient rule with $u = x^2$, $v = x+1$.
$\frac{(x+1)\cdot 2x - x^2 \cdot 1}{(x+1)^2}$. Applying quotient rule with $u = x^2$, $v = x+1$.
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State the formula for the Quotient Rule in calculus.
State the formula for the Quotient Rule in calculus.
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$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}$. Standard derivative formula for quotient of two functions.
$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}$. Standard derivative formula for quotient of two functions.
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Determine $u$ and $v$ for $\frac{2x}{x^3}$ using the Quotient Rule.
Determine $u$ and $v$ for $\frac{2x}{x^3}$ using the Quotient Rule.
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$u = 2x$, $v = x^3$. Linear numerator, cubic denominator.
$u = 2x$, $v = x^3$. Linear numerator, cubic denominator.
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Find the derivative of $\frac{1}{x}$ using the Quotient Rule.
Find the derivative of $\frac{1}{x}$ using the Quotient Rule.
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$\frac{x \cdot 0 - 1 \cdot 1}{x^2}$. Constant numerator means $u' = 0$.
$\frac{x \cdot 0 - 1 \cdot 1}{x^2}$. Constant numerator means $u' = 0$.
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Choose $u$ and $v$ for $\frac{\tan x}{\cos x}$ in the Quotient Rule.
Choose $u$ and $v$ for $\frac{\tan x}{\cos x}$ in the Quotient Rule.
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$u = \tan x$, $v = \cos x$. Numerator and denominator for trigonometric quotient.
$u = \tan x$, $v = \cos x$. Numerator and denominator for trigonometric quotient.
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Find $u'$ for $u = \sin x$ in the Quotient Rule.
Find $u'$ for $u = \sin x$ in the Quotient Rule.
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$u' = \cos x$. Derivative of sine function.
$u' = \cos x$. Derivative of sine function.
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Find $u'$ for $u = \tan x$ in the Quotient Rule.
Find $u'$ for $u = \tan x$ in the Quotient Rule.
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$u' = \sec^2 x$. Derivative of tangent function.
$u' = \sec^2 x$. Derivative of tangent function.
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Determine $u$ and $v$ for $\frac{1}{e^x}$ using the Quotient Rule.
Determine $u$ and $v$ for $\frac{1}{e^x}$ using the Quotient Rule.
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$u = 1$, $v = e^x$. Constant numerator, exponential denominator.
$u = 1$, $v = e^x$. Constant numerator, exponential denominator.
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What is $v'$ if $v = \tan x$ in the Quotient Rule?
What is $v'$ if $v = \tan x$ in the Quotient Rule?
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$v' = \sec^2 x$. Derivative of tangent function.
$v' = \sec^2 x$. Derivative of tangent function.
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What is the derivative of $\frac{\cos x}{x}$ using the Quotient Rule?
What is the derivative of $\frac{\cos x}{x}$ using the Quotient Rule?
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$\frac{x \cdot (-\sin x) - \cos x \cdot 1}{x^2}$. Quotient rule with $u = \cos x$, $v = x$.
$\frac{x \cdot (-\sin x) - \cos x \cdot 1}{x^2}$. Quotient rule with $u = \cos x$, $v = x$.
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Find the derivative of $\frac{3x}{x^2+1}$ using the Quotient Rule.
Find the derivative of $\frac{3x}{x^2+1}$ using the Quotient Rule.
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$\frac{(x^2+1)\cdot 3 - 3x \cdot 2x}{(x^2+1)^2}$. Quotient rule with $u = 3x$, $v = x^2+1$.
$\frac{(x^2+1)\cdot 3 - 3x \cdot 2x}{(x^2+1)^2}$. Quotient rule with $u = 3x$, $v = x^2+1$.
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What is the derivative of $\frac{x^2}{x^2+1}$ using the Quotient Rule?
What is the derivative of $\frac{x^2}{x^2+1}$ using the Quotient Rule?
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$\frac{(x^2+1)\cdot 2x - x^2 \cdot 2x}{(x^2+1)^2}$. Quotient rule applied to rational function.
$\frac{(x^2+1)\cdot 2x - x^2 \cdot 2x}{(x^2+1)^2}$. Quotient rule applied to rational function.
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What is the derivative of $\frac{\ln x}{e^x}$ using the Quotient Rule?
What is the derivative of $\frac{\ln x}{e^x}$ using the Quotient Rule?
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$\frac{e^x \cdot \frac{1}{x} - \ln x \cdot e^x}{(e^x)^2}$. Quotient rule with mixed functions.
$\frac{e^x \cdot \frac{1}{x} - \ln x \cdot e^x}{(e^x)^2}$. Quotient rule with mixed functions.
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Identify $u$ and $v$ for $\frac{x+1}{x^2}$ in the Quotient Rule.
Identify $u$ and $v$ for $\frac{x+1}{x^2}$ in the Quotient Rule.
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$u = x+1$, $v = x^2$. Linear numerator, quadratic denominator.
$u = x+1$, $v = x^2$. Linear numerator, quadratic denominator.
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What is $v'$ if $v = e^x$ in the Quotient Rule?
What is $v'$ if $v = e^x$ in the Quotient Rule?
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$v' = e^x$. Exponential function is its own derivative.
$v' = e^x$. Exponential function is its own derivative.
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Find $u'$ when $u = x^4$ in the Quotient Rule.
Find $u'$ when $u = x^4$ in the Quotient Rule.
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$u' = 4x^3$. Power rule applied to fourth power.
$u' = 4x^3$. Power rule applied to fourth power.
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What is the derivative of $\frac{x^4}{x^2}$ using the Quotient Rule?
What is the derivative of $\frac{x^4}{x^2}$ using the Quotient Rule?
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$\frac{x^2 \cdot 4x^3 - x^4 \cdot 2x}{(x^2)^2}$. Could simplify to $x^2$ first, then derivative is $2x$.
$\frac{x^2 \cdot 4x^3 - x^4 \cdot 2x}{(x^2)^2}$. Could simplify to $x^2$ first, then derivative is $2x$.
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Find the derivative of $\frac{x^3}{3x+2}$ using the Quotient Rule.
Find the derivative of $\frac{x^3}{3x+2}$ using the Quotient Rule.
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$\frac{(3x+2)\cdot 3x^2 - x^3 \cdot 3}{(3x+2)^2}$. Quotient rule with cubic numerator.
$\frac{(3x+2)\cdot 3x^2 - x^3 \cdot 3}{(3x+2)^2}$. Quotient rule with cubic numerator.
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Find $u'$ for $u = x^3$ in the Quotient Rule.
Find $u'$ for $u = x^3$ in the Quotient Rule.
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$u' = 3x^2$. Cubic function derivative.
$u' = 3x^2$. Cubic function derivative.
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Identify $u$ and $v$ for $\frac{x^5}{\ln x}$ in the Quotient Rule.
Identify $u$ and $v$ for $\frac{x^5}{\ln x}$ in the Quotient Rule.
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$u = x^5$, $v = \ln x$. Power and logarithm function identification.
$u = x^5$, $v = \ln x$. Power and logarithm function identification.
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Determine $u$ and $v$ for $\frac{1}{x^2}$ using the Quotient Rule.
Determine $u$ and $v$ for $\frac{1}{x^2}$ using the Quotient Rule.
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$u = 1$, $v = x^2$. Constant numerator, quadratic denominator.
$u = 1$, $v = x^2$. Constant numerator, quadratic denominator.
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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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$\frac{x^2 \cdot 1 - x \cdot 2x}{x^4}$. Quotient rule simplifies to $\frac{1}{x}$ then derivative.
$\frac{x^2 \cdot 1 - x \cdot 2x}{x^4}$. Quotient rule simplifies to $\frac{1}{x}$ then derivative.
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