Question

`\frac{x {}^(2) }{x + y} = y {}^(2) + 1`

I can't figure this out! I need to do implicit differentiation. This is Calculus 1.​

Answer 1

dy/dx = ( 2x - y^2 - 1) / (3y^2 + 2xy + 1).

Explanation:

Cross multiply:

x^2 = (x + y)(y^2 + 1)

Using the Chain and Product rules:

Finding the derivative:

2x = (x + y)(2y dy/dx) + (y^2 + 1)(1 + dy/dx)

2x = 2xy dy/dx + 2y^2 dy/dx + y^2 + y^2 dy/dx + 1 + dy/dx

2xy dy/dx + 2y^2 dy/dx + y^2 dy/dx + dy/dx = 2x - y^2 - 1

3y^2 dy/dx + 2xy dy/dx + dy/dx = 2x - y^2 - 1

dy/dx = ( 2x - y^2 - 1) / (3y^2 + 2xy + 1).