# How do you implicitly differentiate -y=xy-xe^y ?

Jul 5, 2016

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

#### Explanation:

I'm finding $\frac{\mathrm{dy}}{\mathrm{dx}}$. Differentiate term by term, the derivative of y wrt x is $\frac{\mathrm{dy}}{\mathrm{dx}}$ while remembering to be careful with our product rule for both terms on the right hand side.

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

Collect like terms

$\frac{\mathrm{dy}}{\mathrm{dx}} \left[x {e}^{y} - x - 1\right] = y - {e}^{y}$

and solve:

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