# How do you implicitly differentiate 22=(y)/(x-e^y)?

Apr 28, 2017

$\textcolor{red}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{22}{1 + 22 {e}^{y}}}$

#### Explanation:

Implicit differentiation is basically done in cases where $y$ cannot be explicitly written as a function of $x$.

In this case,

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

$\implies 22 \cdot \left(x - {e}^{y}\right) = y$

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

Differentiating both sides with respect to $x$,

$\frac{\mathrm{dx}}{\mathrm{dx}} - \frac{{\mathrm{de}}^{y}}{\mathrm{dx}} = \frac{1}{22} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$

Using chain rule to evaluate $\frac{{\mathrm{de}}^{y}}{\mathrm{dx}}$

$\implies 1 - \frac{{\mathrm{de}}^{y}}{\mathrm{dy}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{22} \frac{\mathrm{dy}}{\mathrm{dx}}$

$\implies 1 - {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{22} \frac{\mathrm{dy}}{\mathrm{dx}}$

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

$\implies \frac{1 + 22 {e}^{y}}{22} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = 1$

$\implies \textcolor{red}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{22}{1 + 22 {e}^{y}}}$