If $xy = e^{x - y}$, what is $\frac{dy}{dx}$?

Explanation

1. 01

   Take logarithms: $\log x + \log y = x - y$.

2. 02

   Differentiate: $\frac{1}{x} + \frac{1}{y}\frac{dy}{dx} = 1 - \frac{dy}{dx}$, so $\frac{dy}{dx}\left(\frac{1}{y} + 1\right) = 1 - \frac{1}{x}$.

3. 03

   $\frac{dy}{dx} = \frac{y(x - 1)}{x(y + 1)}$.