Implicit Differentiation - AP Calculus AB
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Differentiate $x^2 - y^2 = 4$ using implicit differentiation.
Differentiate $x^2 - y^2 = 4$ using implicit differentiation.
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$2x - 2y\frac{dy}{dx} = 0$. Apply power rule to both $x^2$ and $y^2$ terms.
$2x - 2y\frac{dy}{dx} = 0$. Apply power rule to both $x^2$ and $y^2$ terms.
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Differentiate $x^2y^2 = 4$ with respect to $x$.
Differentiate $x^2y^2 = 4$ with respect to $x$.
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$2xy^2 + 2x^2y\frac{dy}{dx} = 0$. Use product rule: $\frac{d}{dx}[x^2y^2] = 2x(y^2) + x^2(2y\frac{dy}{dx})$.
$2xy^2 + 2x^2y\frac{dy}{dx} = 0$. Use product rule: $\frac{d}{dx}[x^2y^2] = 2x(y^2) + x^2(2y\frac{dy}{dx})$.
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What is the first step in implicit differentiation?
What is the first step in implicit differentiation?
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Differentiate both sides with respect to $x$. This sets up the differentiation process.
Differentiate both sides with respect to $x$. This sets up the differentiation process.
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Differentiate $x^2 + 2xy + y^2 = 16$ with respect to $x$.
Differentiate $x^2 + 2xy + y^2 = 16$ with respect to $x$.
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$2x + 2y + 2x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. This is $(x+y)^2 = 16$; use chain rule.
$2x + 2y + 2x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. This is $(x+y)^2 = 16$; use chain rule.
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Which rule is applied when differentiating $y^2$ with respect to $x$?
Which rule is applied when differentiating $y^2$ with respect to $x$?
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The Chain Rule. Since $y$ depends on $x$, we need $\frac{dy}{dx}$ when differentiating $y^2$.
The Chain Rule. Since $y$ depends on $x$, we need $\frac{dy}{dx}$ when differentiating $y^2$.
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Differentiate $x^2y + y^3 = 0$ using implicit differentiation.
Differentiate $x^2y + y^3 = 0$ using implicit differentiation.
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$2xy + x^2\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$. Apply product rule to $x^2y$ and chain rule to $y^3$.
$2xy + x^2\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$. Apply product rule to $x^2y$ and chain rule to $y^3$.
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Find the derivative of $xy = x + y$ using implicit differentiation.
Find the derivative of $xy = x + y$ using implicit differentiation.
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$y + x\frac{dy}{dx} = 1 + \frac{dy}{dx}$. Apply product rule to left side.
$y + x\frac{dy}{dx} = 1 + \frac{dy}{dx}$. Apply product rule to left side.
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Differentiate $x^3 + 3xy + y^3 = 0$ using implicit differentiation.
Differentiate $x^3 + 3xy + y^3 = 0$ using implicit differentiation.
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$3x^2 + 3y + 3x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$. Apply chain rule to $x^3$ and $y^3$, product rule to $3xy$.
$3x^2 + 3y + 3x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0$. Apply chain rule to $x^3$ and $y^3$, product rule to $3xy$.
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Differentiate $2x^2y = 3$ using implicit differentiation.
Differentiate $2x^2y = 3$ using implicit differentiation.
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$4xy + 2x^2\frac{dy}{dx} = 0$. Use product rule on $2x^2y$.
$4xy + 2x^2\frac{dy}{dx} = 0$. Use product rule on $2x^2y$.
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Differentiate $x^2 - y^2 = 4$ using implicit differentiation.
Differentiate $x^2 - y^2 = 4$ using implicit differentiation.
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$2x - 2y\frac{dy}{dx} = 0$. Apply power rule to both $x^2$ and $y^2$ terms.
$2x - 2y\frac{dy}{dx} = 0$. Apply power rule to both $x^2$ and $y^2$ terms.
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Find $\frac{dy}{dx}$ for $y = x^2y + x$.
Find $\frac{dy}{dx}$ for $y = x^2y + x$.
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$\frac{dy}{dx} = 2xy + x^2\frac{dy}{dx} + 1$. Use product rule for $x^2y$ term.
$\frac{dy}{dx} = 2xy + x^2\frac{dy}{dx} + 1$. Use product rule for $x^2y$ term.
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Find the derivative of $x^2y + y^2 = 10$ using implicit differentiation.
Find the derivative of $x^2y + y^2 = 10$ using implicit differentiation.
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$2xy + x^2\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Use product rule for $x^2y$ and power rule for $y^2$.
$2xy + x^2\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Use product rule for $x^2y$ and power rule for $y^2$.
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Differentiate $x + y = xy$ with respect to $x$.
Differentiate $x + y = xy$ with respect to $x$.
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$1 + \frac{dy}{dx} = y + x\frac{dy}{dx}$. Use product rule on right side $xy$.
$1 + \frac{dy}{dx} = y + x\frac{dy}{dx}$. Use product rule on right side $xy$.
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What is the derivative of $xy = 1$ using implicit differentiation?
What is the derivative of $xy = 1$ using implicit differentiation?
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$x\frac{dy}{dx} + y = 0$. Use product rule on $xy$.
$x\frac{dy}{dx} + y = 0$. Use product rule on $xy$.
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Differentiate $x^2 + y^2 = 9$ using implicit differentiation.
Differentiate $x^2 + y^2 = 9$ using implicit differentiation.
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$2x + 2y\frac{dy}{dx} = 0$. Same form as circle equation with radius 3.
$2x + 2y\frac{dy}{dx} = 0$. Same form as circle equation with radius 3.
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Differentiate $x^3 + y^3 = 6xy$ using implicit differentiation.
Differentiate $x^3 + y^3 = 6xy$ using implicit differentiation.
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$3x^2 + 3y^2\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}$. Use chain rule for $y^3$ and product rule for $6xy$.
$3x^2 + 3y^2\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}$. Use chain rule for $y^3$ and product rule for $6xy$.
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Differentiate $\frac{x^3}{y} = 1$ with respect to $x$.
Differentiate $\frac{x^3}{y} = 1$ with respect to $x$.
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$3x^2 - \frac{x^3\frac{dy}{dx}}{y^2} = 0$. Apply quotient rule to $\frac{x^3}{y}$.
$3x^2 - \frac{x^3\frac{dy}{dx}}{y^2} = 0$. Apply quotient rule to $\frac{x^3}{y}$.
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Differentiate: $x^2 + y^2 = 1$ with respect to $x$.
Differentiate: $x^2 + y^2 = 1$ with respect to $x$.
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$2x + 2y\frac{dy}{dx} = 0$. Apply power rule to each term, using chain rule for $y^2$.
$2x + 2y\frac{dy}{dx} = 0$. Apply power rule to each term, using chain rule for $y^2$.
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Differentiate $xy^2 = 4$ with respect to $x$.
Differentiate $xy^2 = 4$ with respect to $x$.
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$y^2 + 2xy\frac{dy}{dx} = 0$. Apply product rule to $xy^2$.
$y^2 + 2xy\frac{dy}{dx} = 0$. Apply product rule to $xy^2$.
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Find the derivative of $x^2 + xy + y^2 = 7$.
Find the derivative of $x^2 + xy + y^2 = 7$.
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$2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Apply product rule to $xy$ and power rule to other terms.
$2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Apply product rule to $xy$ and power rule to other terms.
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What is the result of differentiating $x + y = 1$ implicitly?
What is the result of differentiating $x + y = 1$ implicitly?
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$1 + \frac{dy}{dx} = 0$. Each term differentiates to its coefficient.
$1 + \frac{dy}{dx} = 0$. Each term differentiates to its coefficient.
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Find $\frac{dy}{dx}$ for $y^2 = x^2 + 2$.
Find $\frac{dy}{dx}$ for $y^2 = x^2 + 2$.
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$2y\frac{dy}{dx} = 2x$. Apply power rule to $y^2$ using chain rule.
$2y\frac{dy}{dx} = 2x$. Apply power rule to $y^2$ using chain rule.
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Differentiate $3x^2 + 4y^2 = 12$ with respect to $x$.
Differentiate $3x^2 + 4y^2 = 12$ with respect to $x$.
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$6x + 8y\frac{dy}{dx} = 0$. Apply power rule to each term.
$6x + 8y\frac{dy}{dx} = 0$. Apply power rule to each term.
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Find $\frac{dy}{dx}$ for $y^3 + xy = 1$.
Find $\frac{dy}{dx}$ for $y^3 + xy = 1$.
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$3y^2\frac{dy}{dx} + y + x\frac{dy}{dx} = 0$. Apply chain rule to $y^3$ and product rule to $xy$.
$3y^2\frac{dy}{dx} + y + x\frac{dy}{dx} = 0$. Apply chain rule to $y^3$ and product rule to $xy$.
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Find the derivative of $y$ in $3x^2 + 2xy - y^3 = 0$.
Find the derivative of $y$ in $3x^2 + 2xy - y^3 = 0$.
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$6x + 2y + 2x\frac{dy}{dx} - 3y^2\frac{dy}{dx} = 0$. Use product rule for $2xy$ and chain rule for $y^3$.
$6x + 2y + 2x\frac{dy}{dx} - 3y^2\frac{dy}{dx} = 0$. Use product rule for $2xy$ and chain rule for $y^3$.
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Differentiate $\frac{x}{y} = 1$ with respect to $x$.
Differentiate $\frac{x}{y} = 1$ with respect to $x$.
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$\frac{y - x\frac{dy}{dx}}{y^2} = 0$. Apply quotient rule: $\frac{d}{dx}[\frac{x}{y}] = \frac{y - x\frac{dy}{dx}}{y^2}$.
$\frac{y - x\frac{dy}{dx}}{y^2} = 0$. Apply quotient rule: $\frac{d}{dx}[\frac{x}{y}] = \frac{y - x\frac{dy}{dx}}{y^2}$.
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Differentiate: $\frac{dy}{dx}$ if $x^2 + y^2 = r^2$.
Differentiate: $\frac{dy}{dx}$ if $x^2 + y^2 = r^2$.
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$2x + 2y \frac{dy}{dx} = 0$. Same as differentiating $x^2 + y^2 = 1$ but with constant $r^2$.
$2x + 2y \frac{dy}{dx} = 0$. Same as differentiating $x^2 + y^2 = 1$ but with constant $r^2$.
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Differentiate $3x^2 + 4y^2 = 12$ with respect to $x$.
Differentiate $3x^2 + 4y^2 = 12$ with respect to $x$.
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$6x + 8y \frac{dy}{dx} = 0$. Apply power rule to each term.
$6x + 8y \frac{dy}{dx} = 0$. Apply power rule to each term.
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Differentiate $2x^2y = 3$ using implicit differentiation.
Differentiate $2x^2y = 3$ using implicit differentiation.
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$4xy + 2x^2\frac{dy}{dx} = 0$. Use product rule on $2x^2y$.
$4xy + 2x^2\frac{dy}{dx} = 0$. Use product rule on $2x^2y$.
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What is implicit differentiation?
What is implicit differentiation?
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A technique to find derivatives of equations not solved for $y$. Used when $y$ is not isolated on one side of the equation.
A technique to find derivatives of equations not solved for $y$. Used when $y$ is not isolated on one side of the equation.
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