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Katen [24]
3 years ago
15

Find dy/dx for 4 - xy = y^3

Mathematics
1 answer:
storchak [24]3 years ago
7 0

Answer:

\frac{dy}{dx}=-\frac{y}{3y^2+x}

Step-by-step explanation:

4-xy=y^3

dy/dx=?

\frac{d(4-xy)}{dx}=\frac{d(y^3)}{dx}\\ \frac{d(4)}{dx}-\frac{d(xy)}{dx}=3y^{3-1}\frac{dy}{dx}\\ 0-(\frac{dx}{dx}y+x\frac{dy}{dx})=3y^2\frac{dy}{dx}\\ -(1y+x\frac{dy}{dx})=3y^2\frac{dy}{dx}\\ -(y+x\frac{dy}{dx})=3y^2\frac{dy}{dx}\\ -y-x\frac{dy}{dx}=3y^2\frac{dy}{dx}

Solving for dy/dx: Addind x dy/dx both sides of the equation:

-y-x\frac{dy}{dx}+x\frac{dy}{dx}=3y^2\frac{dy}{dx}+x\frac{dy}{dx} \\ -y=3y^2\frac{dy}{dx}+x\frac{dy}{dx}

Common factor dy/dx on the right side of the equation:

-y=(3y^2+x)\frac{dy}{dx}

Dividing both sides of the equation by 3y^2+x:

\frac{-y}{3y^2+x}=\frac{(3y^2+x)}{3y^2+x}\frac{dy}{dx}\\ -\frac{y}{3y^2+x}=\frac{dy}{dx}\\ \frac{dy}{dx}=-\frac{y}{3y^2+x}

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