# How do you find dy/dx by implicit differentiation given x/y+3y=2?

Feb 28, 2017

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

#### Explanation:

$\frac{x}{y} + 3 y = 2$

$\Rightarrow \frac{x}{y} = 2 - 3 y$

differentiate all terms on both sides $\textcolor{b l u e}{\text{implicitly with respect to x}}$

$\text{differentiate "x/y" using the " color(blue)"quotient rule}$

$\Rightarrow \frac{y .1 - x . \frac{\mathrm{dy}}{\mathrm{dx}}}{{y}^{2}} = 0 - 3. \frac{\mathrm{dy}}{\mathrm{dx}}$

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

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

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