# How do you differentiate y=e^x/(1+x)?

Dec 25, 2016

The answer is $= \frac{x {e}^{x}}{1 + x} ^ 2$

#### Explanation:

We need the differentiation of a quotient

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

Here, $u = {e}^{x}$, $\implies$, $u ' = {e}^{x}$

$v = 1 + x$, $\implies$, $v ' = 1$

Therefore,

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

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

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