Implicit Differentiation - AP Calculus BC
Card 1 of 30
Differentiate $\text{cos}(x + y) = x$ implicitly.
Differentiate $\text{cos}(x + y) = x$ implicitly.
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$-\text{sin}(x+y)(1 + \frac{dy}{dx}) = 1$. Chain rule: $\frac{d}{dx}[\cos(u)] = -\sin(u)\frac{du}{dx}$.
$-\text{sin}(x+y)(1 + \frac{dy}{dx}) = 1$. Chain rule: $\frac{d}{dx}[\cos(u)] = -\sin(u)\frac{du}{dx}$.
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Differentiate $y^3 + 3xy = 6$ implicitly.
Differentiate $y^3 + 3xy = 6$ implicitly.
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$3y^2\frac{dy}{dx} + 3y + 3x\frac{dy}{dx} = 0$. Power rule on $y^3$, product rule on $3xy$ term.
$3y^2\frac{dy}{dx} + 3y + 3x\frac{dy}{dx} = 0$. Power rule on $y^3$, product rule on $3xy$ term.
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Differentiate $y^2 + yx = 1$ implicitly.
Differentiate $y^2 + yx = 1$ implicitly.
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$2y\frac{dy}{dx} + y + x\frac{dy}{dx} = 0$. Product rule on $yx$ term, power rule on $y^2$ term.
$2y\frac{dy}{dx} + y + x\frac{dy}{dx} = 0$. Product rule on $yx$ term, power rule on $y^2$ term.
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Differentiate $x^2y + y^2x = 1$ implicitly.
Differentiate $x^2y + y^2x = 1$ implicitly.
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$2xy + x^2\frac{dy}{dx} + 2yx\frac{dy}{dx} + y^2 = 0$. Use product rule on both terms: $x^2y$ and $y^2x$.
$2xy + x^2\frac{dy}{dx} + 2yx\frac{dy}{dx} + y^2 = 0$. Use product rule on both terms: $x^2y$ and $y^2x$.
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Find $\frac{dy}{dx}$ for $x^2 + y^2 = 1$.
Find $\frac{dy}{dx}$ for $x^2 + y^2 = 1$.
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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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Differentiate $\frac{x}{y} = 1$ implicitly.
Differentiate $\frac{x}{y} = 1$ implicitly.
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$\frac{y - x\frac{dy}{dx}}{y^2} = 0$. Use quotient rule: $\frac{d}{dx}[\frac{u}{v}] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.
$\frac{y - x\frac{dy}{dx}}{y^2} = 0$. Use quotient rule: $\frac{d}{dx}[\frac{u}{v}] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.
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Find $\frac{dy}{dx}$ for $xy = 1$.
Find $\frac{dy}{dx}$ for $xy = 1$.
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$\frac{dy}{dx} = -\frac{y}{x}$. Solve for $\frac{dy}{dx}$ from the differentiated equation.
$\frac{dy}{dx} = -\frac{y}{x}$. Solve for $\frac{dy}{dx}$ from the differentiated equation.
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Differentiate $xy = 1$ implicitly.
Differentiate $xy = 1$ implicitly.
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$y + x\frac{dy}{dx} = 0$. Use product rule: $\frac{d}{dx}(xy) = y + x\frac{dy}{dx}$.
$y + x\frac{dy}{dx} = 0$. Use product rule: $\frac{d}{dx}(xy) = y + x\frac{dy}{dx}$.
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Find $\frac{dy}{dx}$ for $x^2 - xy + y^2 = 7$.
Find $\frac{dy}{dx}$ for $x^2 - xy + y^2 = 7$.
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$\frac{dy}{dx} = \frac{2x - y}{x - 2y}$. Collect $\frac{dy}{dx}$ terms and solve for the derivative.
$\frac{dy}{dx} = \frac{2x - y}{x - 2y}$. Collect $\frac{dy}{dx}$ terms and solve for the derivative.
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Differentiate $\text{ln}(x + y) = x - y$ implicitly.
Differentiate $\text{ln}(x + y) = x - y$ implicitly.
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$\frac{1}{x+y}(1 + \frac{dy}{dx}) = 1 - \frac{dy}{dx}$. Chain rule for $\ln(x+y)$ gives $\frac{1}{x+y}(1 + \frac{dy}{dx})$.
$\frac{1}{x+y}(1 + \frac{dy}{dx}) = 1 - \frac{dy}{dx}$. Chain rule for $\ln(x+y)$ gives $\frac{1}{x+y}(1 + \frac{dy}{dx})$.
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Differentiate $e^{x+y} = xy$ implicitly.
Differentiate $e^{x+y} = xy$ implicitly.
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$e^{x+y}(1+\frac{dy}{dx}) = y + x\frac{dy}{dx}$. Chain rule on left, product rule on right side.
$e^{x+y}(1+\frac{dy}{dx}) = y + x\frac{dy}{dx}$. Chain rule on left, product rule on right side.
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Differentiate $x^2 + y^2 = 4xy$ implicitly.
Differentiate $x^2 + y^2 = 4xy$ implicitly.
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$2x + 2y\frac{dy}{dx} = 4y + 4x\frac{dy}{dx}$. Standard implicit differentiation with product rule on right.
$2x + 2y\frac{dy}{dx} = 4y + 4x\frac{dy}{dx}$. Standard implicit differentiation with product rule on right.
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State the chain rule for differentiation.
State the chain rule for differentiation.
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If $y=f(u)$ and $u=g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Essential for differentiating composite functions.
If $y=f(u)$ and $u=g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Essential for differentiating composite functions.
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Differentiate $e^{xy} = y$ implicitly.
Differentiate $e^{xy} = y$ implicitly.
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$e^{xy}(y + x\frac{dy}{dx}) = \frac{dy}{dx}$. Chain rule on left, product rule within exponent.
$e^{xy}(y + x\frac{dy}{dx}) = \frac{dy}{dx}$. Chain rule on left, product rule within exponent.
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Differentiate $x^2 + y^2 = 1$ implicitly.
Differentiate $x^2 + y^2 = 1$ implicitly.
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$2x + 2y\frac{dy}{dx} = 0$. Apply $\frac{d}{dx}$ to both sides, using chain rule for $y^2$ term.
$2x + 2y\frac{dy}{dx} = 0$. Apply $\frac{d}{dx}$ to both sides, using chain rule for $y^2$ term.
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Find $\frac{dy}{dx}$ for $\frac{1}{x+y} = x-y$.
Find $\frac{dy}{dx}$ for $\frac{1}{x+y} = x-y$.
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$\frac{dy}{dx} = \frac{(x+y)^2 - 1}{1 + (x+y)^2}$. Solve by collecting $\frac{dy}{dx}$ terms on one side.
$\frac{dy}{dx} = \frac{(x+y)^2 - 1}{1 + (x+y)^2}$. Solve by collecting $\frac{dy}{dx}$ terms on one side.
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Find $\frac{dy}{dx}$ for $x^2y + y^2x = 1$.
Find $\frac{dy}{dx}$ for $x^2y + y^2x = 1$.
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$\frac{dy}{dx} = \frac{-2xy - y^2}{x^2 + 2yx}$. Factor out $\frac{dy}{dx}$ from numerator and solve.
$\frac{dy}{dx} = \frac{-2xy - y^2}{x^2 + 2yx}$. Factor out $\frac{dy}{dx}$ from numerator and solve.
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Differentiate $\text{sin}(xy) = y$ implicitly.
Differentiate $\text{sin}(xy) = y$ implicitly.
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$\text{cos}(xy)(y + x\frac{dy}{dx}) = \frac{dy}{dx}$. Chain rule on $\sin(xy)$ with product rule inside.
$\text{cos}(xy)(y + x\frac{dy}{dx}) = \frac{dy}{dx}$. Chain rule on $\sin(xy)$ with product rule inside.
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Find $\frac{dy}{dx}$ for $y^2 + yx = 1$.
Find $\frac{dy}{dx}$ for $y^2 + yx = 1$.
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$\frac{dy}{dx} = \frac{-y}{x + 2y}$. Factor $\frac{dy}{dx}$ and solve for it algebraically.
$\frac{dy}{dx} = \frac{-y}{x + 2y}$. Factor $\frac{dy}{dx}$ and solve for it algebraically.
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Find $\frac{dy}{dx}$ for $x^2 + 2y^2 = 3xy$.
Find $\frac{dy}{dx}$ for $x^2 + 2y^2 = 3xy$.
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$\frac{dy}{dx} = \frac{3y - 2x}{4y - 3x}$. Collect $\frac{dy}{dx}$ terms and solve for the derivative.
$\frac{dy}{dx} = \frac{3y - 2x}{4y - 3x}$. Collect $\frac{dy}{dx}$ terms and solve for the derivative.
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Differentiate $x^3 + y^3 = 6xy$ implicitly.
Differentiate $x^3 + y^3 = 6xy$ implicitly.
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$3x^2 + 3y^2 \frac{dy}{dx} = 6y + 6x \frac{dy}{dx}$. Apply power rule to each cubic term, product rule to $6xy$.
$3x^2 + 3y^2 \frac{dy}{dx} = 6y + 6x \frac{dy}{dx}$. Apply power rule to each cubic term, product rule to $6xy$.
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Find $\frac{dy}{dx}$ for $\frac{x}{y} = 1$.
Find $\frac{dy}{dx}$ for $\frac{x}{y} = 1$.
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$\frac{dy}{dx} = \frac{y}{x}$. From $\frac{x}{y} = 1$, we get $x = y$, so slopes are equal.
$\frac{dy}{dx} = \frac{y}{x}$. From $\frac{x}{y} = 1$, we get $x = y$, so slopes are equal.
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Find $\frac{dy}{dx}$ for $\text{ln}(x + y) = x - y$.
Find $\frac{dy}{dx}$ for $\text{ln}(x + y) = x - y$.
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$\frac{dy}{dx} = \frac{x+y-1}{x+y+1}$. Collect $\frac{dy}{dx}$ terms and solve algebraically.
$\frac{dy}{dx} = \frac{x+y-1}{x+y+1}$. Collect $\frac{dy}{dx}$ terms and solve algebraically.
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Differentiate $\frac{1}{x+y} = x-y$ implicitly.
Differentiate $\frac{1}{x+y} = x-y$ implicitly.
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$-\frac{1}{(x+y)^2}(1 + \frac{dy}{dx}) = 1 - \frac{dy}{dx}$. Use chain rule on $\frac{1}{x+y} = (x+y)^{-1}$.
$-\frac{1}{(x+y)^2}(1 + \frac{dy}{dx}) = 1 - \frac{dy}{dx}$. Use chain rule on $\frac{1}{x+y} = (x+y)^{-1}$.
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Differentiate $x^2 + 2y^2 = 3xy$ implicitly.
Differentiate $x^2 + 2y^2 = 3xy$ implicitly.
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$2x + 4y\frac{dy}{dx} = 3y + 3x\frac{dy}{dx}$. Apply power rule to each term, product rule to $3xy$.
$2x + 4y\frac{dy}{dx} = 3y + 3x\frac{dy}{dx}$. Apply power rule to each term, product rule to $3xy$.
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Differentiate $\text{ln}(xy) = y^2$ implicitly.
Differentiate $\text{ln}(xy) = y^2$ implicitly.
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$\frac{1}{xy}(y + x\frac{dy}{dx}) = 2y\frac{dy}{dx}$. Chain rule on $\ln(xy)$ and product rule within.
$\frac{1}{xy}(y + x\frac{dy}{dx}) = 2y\frac{dy}{dx}$. Chain rule on $\ln(xy)$ and product rule within.
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Find $\frac{dy}{dx}$ for $e^{x+y} = xy$.
Find $\frac{dy}{dx}$ for $e^{x+y} = xy$.
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$\frac{dy}{dx} = \frac{y - e^{x+y}}{e^{x+y} - x}$. Collect $\frac{dy}{dx}$ terms and solve algebraically.
$\frac{dy}{dx} = \frac{y - e^{x+y}}{e^{x+y} - x}$. Collect $\frac{dy}{dx}$ terms and solve algebraically.
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Differentiate $x^2 - xy + y^2 = 7$ implicitly.
Differentiate $x^2 - xy + y^2 = 7$ implicitly.
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$2x - y - x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Apply power rule to $x^2$ and $y^2$, product rule to $xy$.
$2x - y - x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$. Apply power rule to $x^2$ and $y^2$, product rule to $xy$.
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Find $\frac{dy}{dx}$ for $y^2 + yx = 1$.
Find $\frac{dy}{dx}$ for $y^2 + yx = 1$.
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$\frac{dy}{dx} = \frac{-y}{x + 2y}$. Factor $\frac{dy}{dx}$ and solve for it algebraically.
$\frac{dy}{dx} = \frac{-y}{x + 2y}$. Factor $\frac{dy}{dx}$ and solve for it algebraically.
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Differentiate $\frac{1}{x+y} = x-y$ implicitly.
Differentiate $\frac{1}{x+y} = x-y$ implicitly.
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$-\frac{1}{(x+y)^2}(1 + \frac{dy}{dx}) = 1 - \frac{dy}{dx}$. Use chain rule on $\frac{1}{x+y} = (x+y)^{-1}$.
$-\frac{1}{(x+y)^2}(1 + \frac{dy}{dx}) = 1 - \frac{dy}{dx}$. Use chain rule on $\frac{1}{x+y} = (x+y)^{-1}$.
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