# How do you implicitly differentiate  x^2+2xy-y^2+x=3 ?

Jan 18, 2016

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

#### Explanation:

Differentiating with respect to x : using the 'product rule' for term 2xy .

2x + $2 x \frac{d}{\mathrm{dx}} \left(y\right) + y \frac{d}{\mathrm{dx}} \left(2 x\right) - 2 y \frac{\mathrm{dy}}{\mathrm{dx}} + 1 = 0$

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

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

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