If -x^(2)-x-5y^(2)+y^(3)=2y then find (dy)/(dx) in terms of x and y. Answer: (dy)/(dx)=

Differentiate Terms: Differentiate both sides of the equation with respect to x. We will use implicit differentiation because y is a function of x. Differentiate each term on the left side: d/dx(-x^2) = -2x d/dx(-x) = -1 d/dx(-5y^2) = -10y(dy/dx) because we need to use the chain rule for differentiation of y with respect to x. d/dx(y^3) = 3y^2(dy/dx) for the same reason as above. Differentiate the right side: d/dx(2y) = 2(dy/dx)Write Differentiated Equation: Write down the differentiated equation. -2x - 1 - 10y(dy/dx) + 3y^2(dy/dx) = 2(dy/dx)Collect Terms: Collect all dy/dx terms on one side and the remaining terms on the other side. -10y(dy/dx) + 3y^2(dy/dx) - 2(dy/dx) = 2x + 1Factor Out dy/dx: Factor out dy/dx on the left side of the equation. dy/dx(-10y + 3y^2 - 2) = 2x + 1Solve for dy/dx: Solve for dy/dx. dy/dx = (2x + 1) / (-10y + 3y^2 - 2)

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If
-x^(2)-x-5y^(2)+y^(3)=2y then find
(dy)/(dx) in terms of
x and
y.
Answer:
(dy)/(dx)=

If $-x^{2}-x-5 y^{2}+y^{3}=2 y$ 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

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Q. If

-x^{2}-x-5 y^{2}+y^{3}=2 y

then find

\frac{d y}{d x}

in terms of

x

and

y

\newline

Answer:

\frac{d y}{d x}=

Differentiate Terms: Differentiate both sides of the equation with respect to $x$ . We will use implicit differentiation because $y$ is a function of $x$ . Differentiate each term on the left side: $\frac{d}{dx}(-x^2) = -2x$ $\frac{d}{dx}(-x) = -1$ $\frac{d}{dx}(-5y^2) = -10y\frac{dy}{dx}$ because we need to use the chain rule for differentiation of $y$ with respect to $x$ . $\frac{d}{dx}(y^3) = 3y^2\frac{dy}{dx}$ for the same reason as above. Differentiate the right side: $\frac{d}{dx}(2y) = 2\frac{dy}{dx}$
Write Differentiated Equation: Write down the differentiated equation. $\newline$ $-2x - 1 - 10y\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 2\frac{dy}{dx}$
Collect Terms: Collect all $\frac{dy}{dx}$ terms on one side and the remaining terms on the other side. $\newline$ $-10y\left(\frac{dy}{dx}\right) + 3y^2\left(\frac{dy}{dx}\right) - 2\left(\frac{dy}{dx}\right) = 2x + 1$
Factor Out $\frac{dy}{dx}$ : Factor out $\frac{dy}{dx}$ on the left side of the equation. $\newline$ $\frac{dy}{dx}(-10y + 3y^2 - 2) = 2x + 1$
Solve for $\frac{dy}{dx}$ : Solve for $\frac{dy}{dx}$ . $\frac{dy}{dx} = \frac{2x + 1}{-10y + 3y^2 - 2}$