# What is the derivative of (sqrt(x+13)) / (x-4)(root3(2x+1))?

Oct 6, 2016

$f ' \left(x\right) = \frac{2 \sqrt{x + 13}}{3 \left(x - 4\right) \sqrt[3]{{\left(2 x + 1\right)}^{2}}} - \frac{\sqrt{x + 13} \sqrt[3]{2 x + 1}}{{\left(x - 4\right)}^{2}} + \frac{\sqrt[3]{2 x + 1}}{2 \sqrt{x + 13} \left(x - 4\right)}$

#### Explanation:

I would rewrite the expression so that I could exclusively use the Product Rule and Chain Rule.

$f \left(x\right) = {\left(x + 13\right)}^{\frac{1}{2}} {\left(x - 4\right)}^{- 1} {\left(2 x + 1\right)}^{\frac{1}{3}}$

$f ' \left(x\right) = u v w ' + u v ' w + u ' v w$

$u = {\left(x + 13\right)}^{\frac{1}{2}}$
$u ' = \left(\frac{1}{2}\right) {\left(x + 13\right)}^{- \frac{1}{2}}$

$v = {\left(x - 4\right)}^{- 1}$
$v ' = - {\left(x - 4\right)}^{-} 2$

$w = {\left(2 x + 1\right)}^{\frac{1}{3}}$
$w ' = \left(\frac{1}{3}\right) {\left(2 x + 1\right)}^{- \frac{2}{3}} \left(2\right)$

$f ' \left(x\right) = {\left(x + 13\right)}^{\frac{1}{2}} {\left(x - 4\right)}^{- 1} \left(\frac{1}{3}\right) {\left(2 x + 1\right)}^{- \frac{2}{3}} \left(2\right) + {\left(x + 13\right)}^{\frac{1}{2}} \left(- {\left(x - 4\right)}^{-} 2\right) {\left(2 x + 1\right)}^{\frac{1}{3}} + \left(\frac{1}{2}\right) {\left(x + 13\right)}^{- \frac{1}{2}} {\left(x - 4\right)}^{- 1} {\left(2 x + 1\right)}^{\frac{1}{3}}$

Remove the negative exponents

$f ' \left(x\right) = \frac{{\left(x + 13\right)}^{\frac{1}{2}} \left(\frac{1}{3}\right) \left(2\right)}{\left(x - 4\right) {\left(2 x + 1\right)}^{\frac{2}{3}}} + \frac{{\left(x + 13\right)}^{\frac{1}{2}} {\left(2 x + 1\right)}^{\frac{1}{3}}}{- {\left(x - 4\right)}^{2}} + \frac{\left(\frac{1}{2}\right) {\left(2 x + 1\right)}^{\frac{1}{3}}}{{\left(x + 13\right)}^{\frac{1}{2}} {\left(x - 4\right)}^{1}}$

Rearrange numerical constants and remove negatives from denominators

$f ' \left(x\right) = \frac{2 {\left(x + 13\right)}^{\frac{1}{2}}}{3 \left(x - 4\right) {\left(2 x + 1\right)}^{\frac{2}{3}}} - \frac{{\left(x + 13\right)}^{\frac{1}{2}} {\left(2 x + 1\right)}^{\frac{1}{3}}}{{\left(x - 4\right)}^{2}} + \frac{{\left(2 x + 1\right)}^{\frac{1}{3}}}{2 {\left(x + 13\right)}^{\frac{1}{2}} \left(x - 4\right)}$

$f ' \left(x\right) = \frac{2 \sqrt{x + 13}}{3 \left(x - 4\right) \sqrt[3]{{\left(2 x + 1\right)}^{2}}} - \frac{\sqrt{x + 13} \sqrt[3]{2 x + 1}}{{\left(x - 4\right)}^{2}} + \frac{\sqrt[3]{2 x + 1}}{2 \sqrt{x + 13} \left(x - 4\right)}$