# Question #73545

Feb 4, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5}{2} {x}^{- \frac{1}{2}} - \frac{3}{2} {x}^{\frac{1}{2}} - \frac{5}{6} {x}^{\frac{3}{2}}$

#### Explanation:

We could differentiate f(x) using the $\textcolor{b l u e}{\text{product rule}}$

However. if we distribute we can use the 'simpler' $\textcolor{b l u e}{\text{power rule}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\frac{d}{\mathrm{dx}} \left(a {x}^{n}\right) = n a {x}^{n - 1}} \textcolor{w h i t e}{\frac{2}{2}} |}}} \leftarrow \text{ power rule}$

$\Rightarrow y = {x}^{\frac{1}{2}} \left(5 - x - \frac{1}{3} {x}^{2}\right)$

$\textcolor{w h i t e}{\times \times} = 5 {x}^{\frac{1}{2}} - {x}^{\frac{3}{2}} - \frac{1}{3} {x}^{\frac{5}{2}}$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5}{2} {x}^{- \frac{1}{2}} - \frac{3}{2} {x}^{\frac{1}{2}} - \frac{5}{6} {x}^{\frac{3}{2}}$

you may wish to take this further and give the answer with positive exponents .