# How do you differentiate e^(xy) + x^2 - y^2 = 10?

Jul 4, 2016

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

#### Explanation:

${e}^{x y} + {x}^{2} - {y}^{2} = 10 = f \left(x , y\right)$

${f}_{x} = y {e}^{x y} + 2 x$

${f}_{y} = x {e}^{x y} - 2 y$

by implicit function theorem:
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{{f}_{x}}{{f}_{y}} = - \frac{y {e}^{x y} + 2 x}{x {e}^{x y} - 2 y}$