# How do you use the quotient rule to find the derivative of y=x/(x^2+1) ?

Sep 22, 2014

Quotient Rule

$f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2$

Make all of the necessary substitutions

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

$g \left(x\right) = x$
$g ' \left(x\right) = 1$

$h \left(x\right) = {x}^{2} + 1$
$h ' \left(x\right) = 2 x + 0$
$h ' \left(x\right) = 2 x$

${\left(h \left(x\right)\right)}^{2} = {\left({x}^{2} + 1\right)}^{2}$

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

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

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