Find d²y / dx², dy / dx for the curve whose equation is x² + y²_3xy_6x + 5y + 2 = 0 at the point (2,3)

I assume you meant to write

x² + y² - 3xy - 6x + 5y + 2 = 0

Taking y = y(x) and differentiating both sides with respect to x with the power, product, and chain rules gives

2x + 2y dy/dx - 3y - 3x dy/dx - 6 + 5 dy/dx = 0

Solve for dy/dx :

2x - 3y - 6 + (2y - 3x + 5) dy/dx = 0

(2y - 3x + 5) dy/dx = 6 - 2x + 3y

dy/dx = (6 - 2x + 3y) / (2y - 3x + 5)

Now differentiate both sides again. Using the quotient and chain rules,

d²y/dx² = ((2y - 3x + 5) (-2 + 3 dy/dx) - (6 - 2x + 3y) (2 dy/dx - 3)) / (2y - 3x + 5)²

Expand and simplify the numerator:

d²y/dx² = (5y + 8 + (3 - 5x) dy/dx) / (2y - 3x + 5)²

Substitute dy/dx and simplify d²y/dx² :

d²y/dx² = (5y + 8 + (3 - 5x) (6 - 2x + 3y) / (2y - 3x + 5)) / (2y - 3x + 5)²

d²y/dx² = ((5y + 8) (2y - 3x + 5) + (3 - 5x) (6 - 2x + 3y)) / (2y - 3x + 5)³

d²y/dx² = (10y² + 50y - 30xy + 10x²  - 60x + 58) / (2y - 3x + 5)³

answer: i'm great at algebra

~coco

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