# How do you differentiate  (4x − 2) / (x^2 + 1)?

Jun 2, 2016

$\frac{d}{\mathrm{dx}} \left(\frac{4 x - 2}{{x}^{2} + 1}\right) = \frac{- 4 {x}^{2} + 4 x + 4}{{x}^{2} + 1} ^ 2$

#### Explanation:

To differentiate this, use the quotient rule:

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

In this case, $f = 4 x - 2$ and $g = {x}^{2} + 1$. It then follows that ${f}^{'} = 4$ and ${g}^{'} = 2 x$. Plugging all of these expressions into the formula gives

$\frac{4 \left({x}^{2} + 1\right) - \left(4 x - 2\right) \left(2 x\right)}{{x}^{2} + 1} ^ 2 = \frac{- 4 {x}^{2} + 4 x + 4}{{x}^{2} + 1} ^ 2$