Find dy/dx by implicit differentiation.

x^2/x+y=y^2+7

dy/dx = (2x - y² - 7) / (2xy + 3y² + 7)

Step-by-step explanation:

x² / (x + y) = y² + 7

x² = (x + y) (y² + 7)

Take derivative of both sides with respect to x. Use power rule, product rule, and chain rule.

2x = (x + y) (2y dy/dx) + (y² + 7) (1 + dy/dx)

Simplify.

2x = (2xy + 2y²) dy/dx + y² + y² dy/dx + 7 + 7 dy/dx

2x = (2xy + 3y² + 7) dy/dx + y² + 7

2x - y² - 7 = (2xy + 3y² + 7) dy/dx

dy/dx = (2x - y² - 7) / (2xy + 3y² + 7)