# How do you find the derivative of [e^x / (1 - e^x)]?

Sep 7, 2016

$y ' = {e}^{x} / {\left(1 - {e}^{x}\right)}^{2}$.

#### Explanation:

Let $y = {e}^{x} / \left(1 - {e}^{x}\right)$.

To find $y '$, we will use the Quotient Rule , which states, :

$\left(\frac{u}{v}\right) ' = \frac{v u ' - u v '}{v} ^ 2$.

$y = {e}^{x} / \left(1 - {e}^{x}\right)$

$\Rightarrow y ' = \frac{\left(1 - {e}^{x}\right) \left({e}^{x}\right) ' - {e}^{x} \left(1 - {e}^{x}\right) '}{1 - {e}^{x}} ^ 2$.

$= \frac{\left(1 - {e}^{x}\right) \left({e}^{x}\right) - {e}^{x} \left(1 ' - \left({e}^{x}\right) '\right)}{1 - {e}^{x}} ^ 2$.

$= \frac{\left(1 - {e}^{x}\right) {e}^{x} - {e}^{x} \left(0 - {e}^{x}\right)}{1 - {e}^{x}} ^ 2$.

$= \frac{{e}^{x} - {e}^{2 x} + {e}^{2 x}}{1 - {e}^{x}} ^ 2$

$\therefore y ' = {e}^{x} / {\left(1 - {e}^{x}\right)}^{2}$.

Enjoy Maths.!