Find dy/dx by implicit differentiation. x² /x + y = y² + 5

Solution:

x² /(x + y) = y² + 5

x² = (y² + 5)(x + y)

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

2x = (x + y)d(y² + 5)/dx + (y² + 5)d(x + y)/dx

2x = 2(x + y)ydy/dx + (y² + 5)dx/dx + (y² + 5)dy/dx

2x = y² + 5 + ( 2xy + 2y² + y² + 5 )dy/dx

2x - (y² + 5) = (2xy + 3y² + 5)dy/dx

dy/dx = (2x - y² - 5)/(2xy + 3y² + 5)

Find dy/dx by implicit differentiation. x² /x + y = y² + 5

Summary:

dy/dx by implicit differentiation of x² /x + y = y² + 5 is (2x - y² - 5)/(2xy + 3y² + 5)