# How do you use implicit differentiation to find (dy)/(dx) given x^2+xy+y^2=9?

Jan 2, 2017

$y ' = - 2 \frac{x + y}{x + 2 y}$

#### Explanation:

Differentiate the equation with respect to $x$, considering that based on the chain rule:

$\frac{d}{\mathrm{dx}} f \left(y\right) = \frac{\mathrm{df}}{\mathrm{dy}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$

Thus we have:

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

$2 x + x y ' + y + 2 y y ' = 0$

solving for $y '$:

$y ' \left(x + 2 y\right) = - 2 x - y$

$y ' = - 2 \frac{x + y}{x + 2 y}$