# How do you differentiate f(x)=(12sqrt(x) + x^-4 + 5)/(x^(1/8)) using the quotient rule?

Dec 19, 2016

$\frac{\left(\left(\frac{6}{\sqrt{x}}\right) - 4 {x}^{-} 5\right) {x}^{\frac{1}{8}} - \left(12 \sqrt{x} + {x}^{-} 4 + 5\right) \cdot \frac{1}{8} {x}^{- \frac{7}{8}}}{{x}^{\frac{1}{8}}} ^ 2$

#### Explanation:

I would prefer to avoid the quotient rule by rewriting.

$f \left(x\right) = 12 {x}^{\frac{3}{8}} + {x}^{- \frac{33}{8}} + 5 {x}^{- \frac{1}{8}}$

$f ' \left(x\right) = \frac{9}{2} {x}^{- \frac{5}{8}} - \frac{33}{8} {x}^{- \frac{41}{8}} - \frac{5}{8} {x}^{- \frac{9}{8}}$

$= \frac{9}{2 {x}^{\frac{5}{8}}} - \frac{33}{8 {x}^{\frac{41}{8}}} - \frac{5}{8 {x}^{\frac{9}{8}}}$

$= \frac{36 {x}^{\frac{36}{8}} - 33 - 5 {x}^{\frac{32}{8}}}{8 {x}^{\frac{41}{8}}}$

$= \frac{36 {x}^{\frac{9}{2}} - 33 - 5 {x}^{4}}{8 {x}^{\frac{41}{8}}}$