Exploring Behaviors of Implicit Relations - AP Calculus BC
Card 1 of 30
Find $\frac{dy}{dx}$ for $xy = 1$ implicitly.
Find $\frac{dy}{dx}$ for $xy = 1$ implicitly.
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$\frac{dy}{dx} = -\frac{y}{x}$. Use product rule: derivative of $xy$ when product equals constant.
$\frac{dy}{dx} = -\frac{y}{x}$. Use product rule: derivative of $xy$ when product equals constant.
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Find $\frac{dy}{dx}$ for $x^2 - y^2 = 4$ implicitly.
Find $\frac{dy}{dx}$ for $x^2 - y^2 = 4$ implicitly.
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$\frac{dy}{dx} = \frac{x}{y}$. Differentiate both sides and solve for $\frac{dy}{dx}$.
$\frac{dy}{dx} = \frac{x}{y}$. Differentiate both sides and solve for $\frac{dy}{dx}$.
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Determine $\frac{dy}{dx}$ for $x^2y + y^2 = 1$ implicitly.
Determine $\frac{dy}{dx}$ for $x^2y + y^2 = 1$ implicitly.
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$\frac{dy}{dx} = \frac{-2xy}{x^2 + 2y}$. Use product rule on $x^2y$ and chain rule on $y^2$.
$\frac{dy}{dx} = \frac{-2xy}{x^2 + 2y}$. Use product rule on $x^2y$ and chain rule on $y^2$.
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What is an implicit function?
What is an implicit function?
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A function defined by an implicit relation. Function where relationship between variables is given implicitly.
A function defined by an implicit relation. Function where relationship between variables is given implicitly.
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Find $\frac{dy}{dx}$ for $x^2 - y = \sin(x)$ implicitly.
Find $\frac{dy}{dx}$ for $x^2 - y = \sin(x)$ implicitly.
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$\frac{dy}{dx} = \frac{2x - \cos(x)}{1}$. Direct differentiation since $y$ appears linearly.
$\frac{dy}{dx} = \frac{2x - \cos(x)}{1}$. Direct differentiation since $y$ appears linearly.
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What does implicit differentiation involve?
What does implicit differentiation involve?
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Differentiating both sides of an equation with respect to $x$. Treats $y$ as function of $x$, applying chain rule when needed.
Differentiating both sides of an equation with respect to $x$. Treats $y$ as function of $x$, applying chain rule when needed.
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What is the approach for finding vertical tangents in implicit relations?
What is the approach for finding vertical tangents in implicit relations?
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Determine where $\frac{dx}{dy} = 0$. Infinite slope occurs where denominator of $\frac{dy}{dx}$ is zero.
Determine where $\frac{dx}{dy} = 0$. Infinite slope occurs where denominator of $\frac{dy}{dx}$ is zero.
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What rule is used for implicit differentiation of $xy$?
What rule is used for implicit differentiation of $xy$?
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Product rule: $x \frac{dy}{dx} + y$. Derivative of product $xy$ requires both terms.
Product rule: $x \frac{dy}{dx} + y$. Derivative of product $xy$ requires both terms.
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What is the approach for finding horizontal tangents in implicit relations?
What is the approach for finding horizontal tangents in implicit relations?
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Set $\frac{dy}{dx} = 0$ and solve for points. Zero slope occurs where numerator of $\frac{dy}{dx}$ equals zero.
Set $\frac{dy}{dx} = 0$ and solve for points. Zero slope occurs where numerator of $\frac{dy}{dx}$ equals zero.
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What is the chain rule used for in implicit differentiation?
What is the chain rule used for in implicit differentiation?
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To differentiate composite functions. Essential for differentiating functions of $y$ with respect to $x$.
To differentiate composite functions. Essential for differentiating functions of $y$ with respect to $x$.
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What is the definition of a critical point in the context of implicit relations?
What is the definition of a critical point in the context of implicit relations?
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A point where $\frac{dy}{dx}$ is zero or undefined. Points where slope is zero or vertical tangent occurs.
A point where $\frac{dy}{dx}$ is zero or undefined. Points where slope is zero or vertical tangent occurs.
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What is the purpose of implicit differentiation?
What is the purpose of implicit differentiation?
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To find $\frac{dy}{dx}$ for equations not solved for $y$. Allows finding slopes without solving for $y$ explicitly.
To find $\frac{dy}{dx}$ for equations not solved for $y$. Allows finding slopes without solving for $y$ explicitly.
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Determine $\frac{dy}{dx}$ for $x^2y + y = 3x$ implicitly.
Determine $\frac{dy}{dx}$ for $x^2y + y = 3x$ implicitly.
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$\frac{dy}{dx} = \frac{3 - 2xy}{x^2 + 1}$. Apply product rule to $x^2y$ and solve for derivative.
$\frac{dy}{dx} = \frac{3 - 2xy}{x^2 + 1}$. Apply product rule to $x^2y$ and solve for derivative.
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Find $\frac{dy}{dx}$ for $x^2 + xy + y^2 = 7$ implicitly.
Find $\frac{dy}{dx}$ for $x^2 + xy + y^2 = 7$ implicitly.
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$\frac{dy}{dx} = \frac{-2x - y}{x + 2y}$. Apply product rule to $xy$ and chain rule to other terms.
$\frac{dy}{dx} = \frac{-2x - y}{x + 2y}$. Apply product rule to $xy$ and chain rule to other terms.
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What is the formula for differentiating $\sqrt{y}$ implicitly?
What is the formula for differentiating $\sqrt{y}$ implicitly?
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$\frac{1}{2\sqrt{y}} \frac{dy}{dx}$. Chain rule: derivative of $\sqrt{y}$ is $\frac{1}{2\sqrt{y}}$.
$\frac{1}{2\sqrt{y}} \frac{dy}{dx}$. Chain rule: derivative of $\sqrt{y}$ is $\frac{1}{2\sqrt{y}}$.
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What is the derivative of $\sin(y)$ using implicit differentiation?
What is the derivative of $\sin(y)$ using implicit differentiation?
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$\cos(y) \frac{dy}{dx}$. Chain rule applied to sine function with $y$ as argument.
$\cos(y) \frac{dy}{dx}$. Chain rule applied to sine function with $y$ as argument.
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Determine $\frac{dy}{dx}$ for $y^3 = x^3$ implicitly.
Determine $\frac{dy}{dx}$ for $y^3 = x^3$ implicitly.
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$\frac{dy}{dx} = \frac{x^2}{y^2}$. Apply chain rule to both cubic terms and simplify.
$\frac{dy}{dx} = \frac{x^2}{y^2}$. Apply chain rule to both cubic terms and simplify.
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Find $\frac{dy}{dx}$ for $x^3 + 2y^3 = 12$ implicitly.
Find $\frac{dy}{dx}$ for $x^3 + 2y^3 = 12$ implicitly.
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$\frac{dy}{dx} = \frac{-x^2}{2y^2}$. Apply chain rule to cubic terms with different coefficients.
$\frac{dy}{dx} = \frac{-x^2}{2y^2}$. Apply chain rule to cubic terms with different coefficients.
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Find $\frac{dy}{dx}$ for $x^2 + y^2 = 1$ using implicit differentiation.
Find $\frac{dy}{dx}$ for $x^2 + y^2 = 1$ using implicit differentiation.
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$\frac{dy}{dx} = -\frac{x}{y}$. Solve for $\frac{dy}{dx}$ by isolating it algebraically.
$\frac{dy}{dx} = -\frac{x}{y}$. Solve for $\frac{dy}{dx}$ by isolating it algebraically.
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Determine $\frac{dy}{dx}$ for $e^x + e^y = 1$ implicitly.
Determine $\frac{dy}{dx}$ for $e^x + e^y = 1$ implicitly.
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$\frac{dy}{dx} = -e^x e^{-y}$. Apply chain rule to both exponential terms separately.
$\frac{dy}{dx} = -e^x e^{-y}$. Apply chain rule to both exponential terms separately.
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Find $\frac{dy}{dx}$ for $x^2 + 3y^2 = 9$ implicitly.
Find $\frac{dy}{dx}$ for $x^2 + 3y^2 = 9$ implicitly.
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$\frac{dy}{dx} = -\frac{x}{3y}$. Apply chain rule to $3y^2$ term in ellipse equation.
$\frac{dy}{dx} = -\frac{x}{3y}$. Apply chain rule to $3y^2$ term in ellipse equation.
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What is the definition of an implicit relation?
What is the definition of an implicit relation?
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An equation involving multiple variables not solved for one variable. Contrasts with explicit relations where one variable is isolated.
An equation involving multiple variables not solved for one variable. Contrasts with explicit relations where one variable is isolated.
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What is the derivative of $e^{xy}$ using implicit differentiation?
What is the derivative of $e^{xy}$ using implicit differentiation?
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$e^{xy}(y + x \frac{dy}{dx})$. Chain rule on exponential with product rule for $xy$.
$e^{xy}(y + x \frac{dy}{dx})$. Chain rule on exponential with product rule for $xy$.
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What is a point of inflection for an implicit relation?
What is a point of inflection for an implicit relation?
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Where the concavity of the curve changes. Where second derivative changes sign, indicating concavity shift.
Where the concavity of the curve changes. Where second derivative changes sign, indicating concavity shift.
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Determine $\frac{dy}{dx}$ for $x^2y + y^2 = 1$ implicitly.
Determine $\frac{dy}{dx}$ for $x^2y + y^2 = 1$ implicitly.
Tap to reveal answer
$\frac{dy}{dx} = \frac{-2xy}{x^2 + 2y}$. Use product rule on $x^2y$ and chain rule on $y^2$.
$\frac{dy}{dx} = \frac{-2xy}{x^2 + 2y}$. Use product rule on $x^2y$ and chain rule on $y^2$.
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What is the derivative of $\sin(y)$ using implicit differentiation?
What is the derivative of $\sin(y)$ using implicit differentiation?
Tap to reveal answer
$\cos(y) \frac{dy}{dx}$. Chain rule applied to sine function with $y$ as argument.
$\cos(y) \frac{dy}{dx}$. Chain rule applied to sine function with $y$ as argument.
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What does implicit differentiation involve?
What does implicit differentiation involve?
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Differentiating both sides of an equation with respect to $x$. Treats $y$ as function of $x$, applying chain rule when needed.
Differentiating both sides of an equation with respect to $x$. Treats $y$ as function of $x$, applying chain rule when needed.
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What is the purpose of implicit differentiation?
What is the purpose of implicit differentiation?
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To find $\frac{dy}{dx}$ for equations not solved for $y$. Allows finding slopes without solving for $y$ explicitly.
To find $\frac{dy}{dx}$ for equations not solved for $y$. Allows finding slopes without solving for $y$ explicitly.
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What rule is used for implicit differentiation of $xy$?
What rule is used for implicit differentiation of $xy$?
Tap to reveal answer
Product rule: $x \frac{dy}{dx} + y$. Derivative of product $xy$ requires both terms.
Product rule: $x \frac{dy}{dx} + y$. Derivative of product $xy$ requires both terms.
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Determine $\frac{dy}{dx}$ for $e^x + e^y = 1$ implicitly.
Determine $\frac{dy}{dx}$ for $e^x + e^y = 1$ implicitly.
Tap to reveal answer
$\frac{dy}{dx} = -e^x e^{-y}$. Apply chain rule to both exponential terms separately.
$\frac{dy}{dx} = -e^x e^{-y}$. Apply chain rule to both exponential terms separately.
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