differentiate. y = e^x / (1 + x). y =

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = e^{x}$, $v=1 + x$.

Step2: Find $u'$ and $v'$

The derivative of $u = e^{x}$ is $u'=e^{x}$, and the derivative of $v = 1 + x$ is $v'=1$.

Step3: Apply quotient - rule

$y'=\frac{e^{x}(1 + x)-e^{x}\times1}{(1 + x)^{2}}$.

Step4: Simplify the expression

$y'=\frac{e^{x}+xe^{x}-e^{x}}{(1 + x)^{2}}=\frac{xe^{x}}{(1 + x)^{2}}$.

Answer:

$\frac{xe^{x}}{(1 + x)^{2}}$