Implicitly Defined Functions - AP Precalculus
Card 1 of 30
Differentiate $x^2 + 2xy + y^2 = 0$ implicitly.
Differentiate $x^2 + 2xy + y^2 = 0$ implicitly.
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$2x + 2y + 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Perfect square form requires product rule on $2xy$.
$2x + 2y + 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Perfect square form requires product rule on $2xy$.
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Differentiate $xy = 4$ implicitly.
Differentiate $xy = 4$ implicitly.
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$y + x \frac{dy}{dx} = 0$. Apply the product rule to the $xy$ term.
$y + x \frac{dy}{dx} = 0$. Apply the product rule to the $xy$ term.
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Differentiate $x^2y + y^3 = 7$ implicitly.
Differentiate $x^2y + y^3 = 7$ implicitly.
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$2xy + x^2 \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Apply the product rule and chain rule to each term.
$2xy + x^2 \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Apply the product rule and chain rule to each term.
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Find $dy/dx$ for $x^3 + y^3 = 6xy$.
Find $dy/dx$ for $x^3 + y^3 = 6xy$.
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$\frac{2y - 3x^2}{3y^2 - 2x}$. Use implicit differentiation on both sides of the equation.
$\frac{2y - 3x^2}{3y^2 - 2x}$. Use implicit differentiation on both sides of the equation.
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Differentiate $3x + 4y = 12$ implicitly.
Differentiate $3x + 4y = 12$ implicitly.
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$3 + 4\frac{dy}{dx} = 0$. Simple linear equation with constant coefficients.
$3 + 4\frac{dy}{dx} = 0$. Simple linear equation with constant coefficients.
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Identify the implicit function in $x^2 + y^2 = 1$.
Identify the implicit function in $x^2 + y^2 = 1$.
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$x^2 + y^2 = 1$. The entire equation is implicit as written.
$x^2 + y^2 = 1$. The entire equation is implicit as written.
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Identify the implicit function in $x^2 + y^2 = 1$.
Identify the implicit function in $x^2 + y^2 = 1$.
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$x^2 + y^2 = 1$. The entire equation is implicit as written.
$x^2 + y^2 = 1$. The entire equation is implicit as written.
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Find the implicit derivative of $x^2 + xy + y^2 = 7$.
Find the implicit derivative of $x^2 + xy + y^2 = 7$.
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$2x + y + x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Use product rule on $xy$ and chain rule on $y^2$.
$2x + y + x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Use product rule on $xy$ and chain rule on $y^2$.
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What is the implicit derivative of $e^x + y = 5$?
What is the implicit derivative of $e^x + y = 5$?
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$e^x + \frac{dy}{dx} = 0$. Simple linear equation with exponential and linear terms.
$e^x + \frac{dy}{dx} = 0$. Simple linear equation with exponential and linear terms.
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Differentiate $x^2 + y^2 = 9$ implicitly.
Differentiate $x^2 + y^2 = 9$ implicitly.
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$2x + 2y \frac{dy}{dx} = 0$. Basic circle equation requiring chain rule for $y^2$.
$2x + 2y \frac{dy}{dx} = 0$. Basic circle equation requiring chain rule for $y^2$.
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What is the implicit derivative of $x^2 + 2y^2 = 1$?
What is the implicit derivative of $x^2 + 2y^2 = 1$?
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$2x + 4y \frac{dy}{dx} = 0$. Chain rule applies to $2y^2$ with coefficient 4.
$2x + 4y \frac{dy}{dx} = 0$. Chain rule applies to $2y^2$ with coefficient 4.
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Differentiate $sin(x) + cos(y) = 1$ implicitly.
Differentiate $sin(x) + cos(y) = 1$ implicitly.
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$cos(x) - sin(y) \frac{dy}{dx} = 0$. Apply chain rule to trigonometric functions of $x$ and $y$.
$cos(x) - sin(y) \frac{dy}{dx} = 0$. Apply chain rule to trigonometric functions of $x$ and $y$.
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Find the implicit derivative for $x^2 + 3xy + 2y^2 = 0$.
Find the implicit derivative for $x^2 + 3xy + 2y^2 = 0$.
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$2x + 3y + 3x \frac{dy}{dx} + 4y \frac{dy}{dx} = 0$. Mixed terms require product rule and chain rule.
$2x + 3y + 3x \frac{dy}{dx} + 4y \frac{dy}{dx} = 0$. Mixed terms require product rule and chain rule.
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What is the chain rule in terms of implicit differentiation?
What is the chain rule in terms of implicit differentiation?
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Use $\frac{dy}{dx}$ for terms of $y$ when differentiating with respect to $x$. Apply $\frac{dy}{dx}$ to all $y$ terms when differentiating.
Use $\frac{dy}{dx}$ for terms of $y$ when differentiating with respect to $x$. Apply $\frac{dy}{dx}$ to all $y$ terms when differentiating.
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Differentiate $x^2y - y = 3$ implicitly.
Differentiate $x^2y - y = 3$ implicitly.
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$2xy + x^2 \frac{dy}{dx} - \frac{dy}{dx} = 0$. Use product rule on $x^2y$ and chain rule on $y$.
$2xy + x^2 \frac{dy}{dx} - \frac{dy}{dx} = 0$. Use product rule on $x^2y$ and chain rule on $y$.
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Differentiate $3x + 4y = 12$ implicitly.
Differentiate $3x + 4y = 12$ implicitly.
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$3 + 4\frac{dy}{dx} = 0$. Simple linear equation with constant coefficients.
$3 + 4\frac{dy}{dx} = 0$. Simple linear equation with constant coefficients.
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Differentiate $x^2 + 2xy + y^2 = 0$ implicitly.
Differentiate $x^2 + 2xy + y^2 = 0$ implicitly.
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$2x + 2y + 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Perfect square form requires product rule on $2xy$.
$2x + 2y + 2x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0$. Perfect square form requires product rule on $2xy$.
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Differentiate implicitly $x^3 + y^3 = 3xy$.
Differentiate implicitly $x^3 + y^3 = 3xy$.
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$3x^2 + 3y^2 \frac{dy}{dx} = 3y + 3x \frac{dy}{dx}$. Apply chain rule to $y^3$ and product rule to $3xy$.
$3x^2 + 3y^2 \frac{dy}{dx} = 3y + 3x \frac{dy}{dx}$. Apply chain rule to $y^3$ and product rule to $3xy$.
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What is the result of implicit differentiation for $y^2 - x^2 = 1$?
What is the result of implicit differentiation for $y^2 - x^2 = 1$?
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$2y \frac{dy}{dx} - 2x = 0$. Differentiate each term using appropriate rules.
$2y \frac{dy}{dx} - 2x = 0$. Differentiate each term using appropriate rules.
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Identify the term to differentiate implicitly: $x^2 + xy + y = 1$.
Identify the term to differentiate implicitly: $x^2 + xy + y = 1$.
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$xy$. The product term requires the product rule for differentiation.
$xy$. The product term requires the product rule for differentiation.
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What does it mean to differentiate implicitly?
What does it mean to differentiate implicitly?
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Differentiate an equation involving multiple variables. Take derivatives without solving for one variable first.
Differentiate an equation involving multiple variables. Take derivatives without solving for one variable first.
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How do you solve for $dy/dx$ in an implicit equation?
How do you solve for $dy/dx$ in an implicit equation?
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Use implicit differentiation, solve for $\frac{dy}{dx}$. Collect terms with $\frac{dy}{dx}$ and factor them out.
Use implicit differentiation, solve for $\frac{dy}{dx}$. Collect terms with $\frac{dy}{dx}$ and factor them out.
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What does it mean to implicitly differentiate $y^2 = 4x$?
What does it mean to implicitly differentiate $y^2 = 4x$?
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Find $\frac{dy}{dx}$ for $y^2 = 4x$. Apply chain rule to find the derivative relationship.
Find $\frac{dy}{dx}$ for $y^2 = 4x$. Apply chain rule to find the derivative relationship.
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Differentiate $x^2y + y^3 = 7$ implicitly.
Differentiate $x^2y + y^3 = 7$ implicitly.
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$2xy + x^2 \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Apply the product rule and chain rule to each term.
$2xy + x^2 \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Apply the product rule and chain rule to each term.
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What is the chain rule for implicit differentiation?
What is the chain rule for implicit differentiation?
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Differentiate both sides with respect to $x$, apply $\frac{dy}{dx}$ to $y$ terms. Treat $y$ as a function of $x$ when differentiating.
Differentiate both sides with respect to $x$, apply $\frac{dy}{dx}$ to $y$ terms. Treat $y$ as a function of $x$ when differentiating.
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Find $dy/dx$ for $x^3 + y^3 = 6xy$.
Find $dy/dx$ for $x^3 + y^3 = 6xy$.
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$\frac{2y - 3x^2}{3y^2 - 2x}$. Use implicit differentiation on both sides of the equation.
$\frac{2y - 3x^2}{3y^2 - 2x}$. Use implicit differentiation on both sides of the equation.
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What is the derivative of $y$ with respect to $x$ for $x^2 + y^2 = 1$?
What is the derivative of $y$ with respect to $x$ for $x^2 + y^2 = 1$?
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$-\frac{x}{y}$. Apply implicit differentiation to the circle equation.
$-\frac{x}{y}$. Apply implicit differentiation to the circle equation.
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What is an implicitly defined function?
What is an implicitly defined function?
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A function defined by an equation not solved for one variable. The equation contains both variables without solving for one.
A function defined by an equation not solved for one variable. The equation contains both variables without solving for one.
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Find the implicit derivative for $x^3 + 3xy + y^3 = 0$.
Find the implicit derivative for $x^3 + 3xy + y^3 = 0$.
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$3x^2 + 3y + 3x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Use product rule on $3xy$ and chain rule on cubic terms.
$3x^2 + 3y + 3x \frac{dy}{dx} + 3y^2 \frac{dy}{dx} = 0$. Use product rule on $3xy$ and chain rule on cubic terms.
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Differentiate $x^2 + y = 2xy$ implicitly.
Differentiate $x^2 + y = 2xy$ implicitly.
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$2x + \frac{dy}{dx} = 2y + 2x \frac{dy}{dx}$. Differentiate each term and apply product rule to $2xy$.
$2x + \frac{dy}{dx} = 2y + 2x \frac{dy}{dx}$. Differentiate each term and apply product rule to $2xy$.
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