# What is the derivative of y=(x^2 - 1) /( x+1)?

Jan 26, 2016

$\frac{2}{{x}^{2} - 1}$

#### Explanation:

Use the quotient rule, which states that if $f \left(x\right) = g \frac{x}{h \left(x\right)}$ then
$f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - h ' \left(x\right) g \left(x\right)}{{g}^{2} \left(x\right)}$

$g \left(x\right) = \left({x}^{2} - 1\right)$ so $g ' \left(x\right) = 2 x$
$h \left(x\right) = \left(x + 1\right)$ so $h ' \left(x\right) = 1$

$f ' \left(x\right) = \frac{\left(x + 1\right) \left(2 x\right) - 1 \cdot \left(x + 1\right)}{{x}^{2} - 1} ^ 2$
$= \frac{2 {x}^{2} + x - 1 - x - 1}{{x}^{2} - 1} ^ 2$
$= \frac{2 {x}^{2} - 2}{{x}^{2} - 1} ^ 2$
$= \frac{2 \left({x}^{2} - 1\right)}{{x}^{2} - 1} ^ 2$
$= \frac{2}{{x}^{2} - 1}$