# How do you derive y = (x^2+8x+3)/x^(1/2) using the quotient rule?

Mar 11, 2018

$\frac{3 {x}^{2} + 8 x - 3}{2 {x}^{\frac{3}{2}}}$

#### Explanation:

$= \frac{\left({x}^{2} + 8 x + 3\right) ' \left({x}^{\frac{1}{2}}\right) - \left({x}^{\frac{1}{2}}\right) ' \left({x}^{2} + 8 x + 3\right)}{{x}^{\frac{1}{2}}} ^ 2$
$= \frac{{x}^{\frac{1}{2}} \left(2 x + 8\right) - \frac{1}{2} {x}^{- \frac{1}{2}} \left({x}^{2} + 8 x + 3\right)}{x}$
$= \frac{2 {x}^{\frac{3}{2}} + 8 {x}^{\frac{1}{2}} - \frac{1}{2} {x}^{\frac{3}{2}} - 4 {x}^{\frac{1}{2}} - \frac{3}{2} {x}^{- \frac{1}{2}}}{x}$
$= \frac{\frac{3}{2} {x}^{\frac{3}{2}} + 4 {x}^{\frac{1}{2}} - \frac{3}{2} {x}^{- \frac{1}{2}}}{x}$
$= \frac{{x}^{- \frac{1}{2}} \left[\frac{3}{2} {x}^{2} + 4 x - \frac{3}{2}\right]}{x}$
$= \frac{\frac{3 {x}^{2} + 8 x + 3}{2}}{x} ^ \left(\frac{3}{2}\right)$
$= \frac{3 {x}^{2} + 8 x + 3}{2 {x}^{\frac{3}{2}}}$