# How do you find dy/dx by implicit differentiation?

## e^(x/y)=3x-3y

Nov 4, 2016

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

#### Explanation:

${e}^{\frac{x}{y}} = 3 x - 3 y$

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

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

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

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

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

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