# How do you implicitly differentiate  3x^2 - 2xy + y^2 = 11 ?

Feb 14, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 x - y}{x - y}$

#### Explanation:

Start by taking the derivative on both sides of the equation with respect to $x$, bearing in mind that $y$ is a function of $x$.
Then solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$.

$\therefore \frac{d}{\mathrm{dx}} \left(3 {x}^{2} - 2 x y + {y}^{2}\right) = \frac{d}{\mathrm{dx}} 11$

$\therefore 6 x - 2 x \frac{\mathrm{dy}}{\mathrm{dx}} - 2 y + 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x - 2 y}{2 x - 2 y}$

$= \frac{2 \left(3 x - y\right)}{2 \left(x - y\right)}$