# How do you differentiate 1=e^(xy)/(e^x-ye^y)?

Apr 1, 2017

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

#### Explanation:

Write the equation as:

${e}^{x} - y {e}^{y} = {e}^{x y}$

Differentiate both sides with respect to $x$:

${e}^{x} - y ' {e}^{y} - y y ' {e}^{y} = {e}^{x y} \left(y + x y '\right)$

Solving for $y '$:

$y ' \left(x {e}^{x y} + {e}^{y} + y {e}^{y}\right) = {e}^{x} - y {e}^{x y}$

$y ' = \frac{{e}^{x} - y {e}^{x y}}{x {e}^{x y} + {e}^{y} \left(1 + y\right)}$