# How do you use implicit differentiation to find dy/dx given x^2+4y^2=7+3xy?

Oct 19, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 y - 2 x}{8 y - 3 x}$

#### Explanation:

${x}^{2} + 4 {y}^{2} = 7 + 3 x y$
$\Rightarrow 2 x + 8 y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = 0 + 3 \left(y + x \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\right)$
$\Rightarrow 2 x + 8 y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = 3 y + 3 x \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$

$\Rightarrow 8 y \frac{\mathrm{dy}}{\mathrm{dx}} - 3 x \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = 3 y - 2 x$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} \left(8 y - 3 x\right) = 3 y - 2 x$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 y - 2 x}{8 y - 3 x}$