# How do you implicitly differentiate 11=(x+y)/(xe^y+ye^x)?

$\frac{\mathrm{dy}}{\mathrm{dx}} = \setminus \frac{11 y {e}^{x} + 11 {e}^{y} - 1}{1 - 11 x {e}^{y} - 11 {e}^{x}}$

#### Explanation:

Given function:

$11 = \frac{x + y}{x {e}^{y} + y {e}^{x}}$

$11 x {e}^{y} + 11 y {e}^{x} = x + y$

Differentiating above equation w.r.t. $x$ on both the sides as follows

$\frac{d}{\mathrm{dx}} \left(11 x {e}^{y} + 11 y {e}^{x}\right) = \frac{d}{\mathrm{dx}} \left(x + y\right)$

$11 x {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} + 11 {e}^{y} + 11 y {e}^{x} + 11 {e}^{x} \frac{\mathrm{dy}}{\mathrm{dx}} = 1 + \frac{\mathrm{dy}}{\mathrm{dx}}$

$\left(1 - 11 x {e}^{y} - 11 {e}^{x}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 11 y {e}^{x} + 11 {e}^{y} - 1$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \setminus \frac{11 y {e}^{x} + 11 {e}^{y} - 1}{1 - 11 x {e}^{y} - 11 {e}^{x}}$