# How do you find the derivative of xe^y=y-1?

Feb 19, 2015

Use implicit differentiation. On the left hand side use the product rule.

We are differentiating with respect to x

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

get everything with $\frac{\mathrm{dy}}{\mathrm{dx}}$ on one side of the equation

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

Factor left hand side

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

now divide both sides by $x {e}^{y} - 1$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {e}^{y} / \left(x {e}^{y} - 1\right)$