# How do you differentiate x^2/(sqrt(x+2))-sqrt(x+2)/x^2?

Aug 13, 2017

#### Explanation:

Let $y = {x}^{2} / \left(\sqrt{x + 2}\right) - \frac{\sqrt{x + 2}}{x} ^ 2$

Note that with $u = {x}^{2} / \sqrt{x + 2}$, we have

$y = u - \frac{1}{u}$.

So $y ' = u ' + \frac{1}{u} ^ 2 u ' = u ' \left(1 + \frac{1}{u} ^ 2\right)$

Use the quotient rule to find

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

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

$= \frac{3 {x}^{2} + 8 x}{2 {\left(x + 2\right)}^{\frac{3}{2}}}$

So

$y ' = \frac{3 {x}^{2} + 8 x}{2 {\left(x + 2\right)}^{\frac{3}{2}}} \left(1 + \frac{1}{{x}^{2} / \sqrt{x + 2}} ^ 2\right)$

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

Simplify further to taste.