# How do you implicitly differentiate 22=(y)/(1-xe^y)?

Apr 7, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{22 {e}^{y}}{y - 23}$

#### Explanation:

We have;

$22 = \frac{y}{1 - x {e}^{y}}$

$\implies 22 - 22 x {e}^{y} = y$

$22 - y = 22 x {e}^{y}$

$\frac{22 - y}{22 {e}^{y}} = x$

$x = \frac{22 - y}{22 {e}^{y}}$

Diff.w.r.t. $\textcolor{red}{y}$ and "using"color(blue)" Quotient Rule"

$\frac{\mathrm{dx}}{\mathrm{dy}} = \frac{22 {e}^{y} \frac{d}{\mathrm{dy}} \left(22 - y\right) - \left(22 - y\right) \frac{d}{\mathrm{dy}} \left(22 {e}^{y}\right)}{22 {e}^{y}} ^ 2$

$\implies \frac{\mathrm{dx}}{\mathrm{dy}} = \frac{22 {e}^{y} \left(- 1\right) - \left(22 - y\right) \left(22 {e}^{y}\right)}{22 {e}^{y}} ^ 2$

$\implies \frac{\mathrm{dx}}{\mathrm{dy}} = \frac{22 {e}^{y} \left(- 1 - 22 + y\right)}{22 {e}^{y}} ^ 2$

$\implies \textcolor{red}{\frac{\mathrm{dx}}{\mathrm{dy}}} = \frac{y - 23}{22 {e}^{y}} \to$,But , $\left[\frac{\mathrm{dx}}{\mathrm{dy}} = \frac{1}{\frac{\mathrm{dy}}{\mathrm{dx}}}\right]$

$\implies \textcolor{red}{\frac{\mathrm{dy}}{\mathrm{dx}}} = \frac{22 {e}^{y}}{y - 23}$