# How do you differentiate 2xy+y^2=x+y?

Aug 14, 2015

2xy + ${y}^{2}$ = x + y

By transpose we get

2xy + ${y}^{2}$ - x - y = 0

Use product rule to differentiate 2xy

2x . $\frac{\mathrm{dy}}{\mathrm{dx}}$ + 2y . 1 + 2y $\frac{\mathrm{dy}}{\mathrm{dx}}$ - 1 - $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 0

2x . $\frac{\mathrm{dy}}{\mathrm{dx}}$ + 2y + 2y $\frac{\mathrm{dy}}{\mathrm{dx}}$ - 1 - $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 0

Rearrange
2x . $\frac{\mathrm{dy}}{\mathrm{dx}}$ + 2y $\frac{\mathrm{dy}}{\mathrm{dx}}$ - $\frac{\mathrm{dy}}{\mathrm{dx}}$ + 2y - 1 = 0

2x . $\frac{\mathrm{dy}}{\mathrm{dx}}$ + 2y $\frac{\mathrm{dy}}{\mathrm{dx}}$ - $\frac{\mathrm{dy}}{\mathrm{dx}}$ = - 2y + 1

$\frac{\mathrm{dy}}{\mathrm{dx}}$ (2x + 2y - 1) = - 2y + 1

$\frac{\mathrm{dy}}{\mathrm{dx}}$ = $\frac{- 2 y + 1}{2 x + 2 y - 1}$