Find dy/dx by implicit differentiation. x² /x + y = y² + 9

Solution:

x² /(x + y) = y² + 9

x² = (y² + 9)(x + y)

Differentiating the above w.r.t. x we have:

2x = (x + y)d(y² + 9)/dx + (y² + 9)d(x + y)/dx

2x = 2(x + y)ydy/dx + (y² + 9)dx/dx + (y² + 9)dy/dx

2x = y² + 9 + ( 2xy + 2y² + y² + 9 )dy/dx

2x - (y² + 9) = (2xy + 3y² + 9)dy/dx

dy/dx = (2x - y² - 9)/(2xy + 3y² + 9)

Find dy/dx by implicit differentiation. x² /x + y = y² + 9

Summary:

dy/dx by implicit differentiation of x² /x + y = y² + 9 is (2x - y² - 9)/(2xy + 3y² + 9)