If -3xy^(2)+x^(3)=-y^(3) then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Apply Product Rule: To find the derivative (dy)/(dx), we need to differentiate both sides of the equation with respect to x, treating y as a function of x (implicit differentiation). Differentiate the left side: -3xy^2 + x^3. Using the product rule for -3xy^2: d/dx(-3xy^2) = -3y^2 * d/dx(x) + -3x * d/dx(y^2). Differentiate x^3: d/dx(x^3) = 3x^2.Apply Chain Rule: Now differentiate the right side: -y^3. Using the chain rule: d/dx(-y^3) = -3y^2 * d/dx(y).Combine Derivatives: Combine the derivatives from both sides to form an equation: -3y^2 * 1 + -3x * 2y * (dy/dx) + 3x^2 = -3y^2 * (dy/dx).Simplify and Solve: Simplify the equation and solve for (dy/dx): -3y^2 - 6xy * (dy/dx) + 3x^2 = -3y^2 * (dy/dx).Rearrange Terms: Move all terms involving (dy/dx) to one side and the rest to the other side: 6xy * (dy/dx) - 3y^2 * (dy/dx) = 3x^2 - 3y^2.Factor Out: Factor out (dy/dx) from the left side: (dy/dx) * (6xy - 3y^2) = 3x^2 - 3y^2.Isolate (dy/dx): Divide both sides by (6xy - 3y^2) to isolate (dy/dx): (dy/dx) = (3x^2 - 3y^2) / (6xy - 3y^2).Simplify Expression: Simplify the expression by factoring out common terms: (dy/dx) = 3(x^2 - y^2) / (3y(2x - y)).Cancel Common Factor: Cancel the common factor of 3: (dy/dx) = (x^2 - y^2) / (y(2x - y)).

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If
-3xy^(2)+x^(3)=-y^(3) then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $-3 x y^{2}+x^{3}=-y^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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Algebra 1

Transformations of quadratic functions

Full solution

Q. If

-3 x y^{2}+x^{3}=-y^{3}

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Apply Product Rule: To find the derivative $\frac{dy}{dx}$ , we need to differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ (implicit differentiation). $\newline$ Differentiate the left side: $-3xy^2 + x^3$ . $\newline$ Using the product rule for $-3xy^2$ : $\frac{d}{dx}(-3xy^2) = -3y^2 \cdot \frac{d}{dx}(x) + -3x \cdot \frac{d}{dx}(y^2)$ . $\newline$ Differentiate $x^3$ : $\frac{d}{dx}(x^3) = 3x^2$ .
Apply Chain Rule: Now differentiate the right side: $-y^3$ . Using the chain rule: $\frac{d}{dx}(-y^3) = -3y^2 \cdot \frac{d}{dx}(y)$ .
Combine Derivatives: Combine the derivatives from both sides to form an equation: $\newline$ $-3y^2 \cdot 1 + -3x \cdot 2y \cdot \left(\frac{dy}{dx}\right) + 3x^2 = -3y^2 \cdot \left(\frac{dy}{dx}\right)$ .
Simplify and Solve: Simplify the equation and solve for $\frac{dy}{dx}$ : $-3y^2 - 6xy \cdot \frac{dy}{dx} + 3x^2 = -3y^2 \cdot \frac{dy}{dx}$ .
Rearrange Terms: Move all terms involving $\frac{dy}{dx}$ to one side and the rest to the other side: $\newline$ $6xy \cdot \frac{dy}{dx} - 3y^2 \cdot \frac{dy}{dx} = 3x^2 - 3y^2$ .
Factor Out: Factor out $(\frac{dy}{dx})$ from the left side: $\newline$ $(\frac{dy}{dx}) \cdot (6xy - 3y^2) = 3x^2 - 3y^2$ .
Isolate $(\frac{dy}{dx}):</b> Divide both sides by \$(6xy - 3y^2)$ to isolate $(\frac{dy}{dx}):$ $\newline$ $(\frac{dy}{dx}) = \frac{3x^2 - 3y^2}{6xy - 3y^2}.$
Simplify Expression: Simplify the expression by factoring out common terms: $(\frac{dy}{dx}) = \frac{3(x^2 - y^2)}{3y(2x - y)}$ .
Cancel Common Factor: Cancel the common factor of $3$ : $(\frac{dy}{dx}) = \frac{x^2 - y^2}{y(2x - y)}$ .

If $-3 x y^{2}+x^{3}=-y^{3}$ then find $\frac{d y}{d x}$ in terms of $x$ and $y$ . $\newline$ Answer: $\frac{d y}{d x}=$

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