find dy/dx
(-x^2)(y)+2xy=3y^2-7
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To find dy/dx, we will use implicit differentiation. Let's go step-by-step:
1. Start with the given equation:
(-x^2)(y) + 2xy = 3y^2 - 7
2. Differentiate both sides of the equation with respect to x. Treat y as a function of x and apply the product rule for differentiation:
d/dx [(-x^2)(y)] + d/dx [2xy] = d/dx [3y^2 - 7]
3. Apply the product rule on the left side:
(-x^2)(dy/dx) + (-2x)(y) + 2x(dy/dx) + 2y = 6y(dy/dx)
4. Combine like terms:
-x^2(dy/dx) - 2xy + 2x(dy/dx) + 2y = 6y(dy/dx)
5. Rearrange the terms:
(2x - x^2)(dy/dx) + (2y - 2xy) = 6y(dy/dx)
6. Now we can solve for dy/dx. First, move all dy/dx terms to one side of the equation:
(2x - x^2 - 6y)(dy/dx) = 2xy - 2y
7. Divide both sides of the equation by (2x - x^2 - 6y) to isolate dy/dx:
dy/dx = (2xy - 2y)/(2x - x^2 - 6y)
Therefore, the equation for dy/dx is dy/dx = (2xy - 2y)/(2x - x^2 - 6y).