# What is the second derivative of  (x^2 + 1/x)^5?

Jun 2, 2018

$f ' ' \left(x\right) = 20 {\left({x}^{2} + \frac{1}{x}\right)}^{3} {\left(2 x - \frac{1}{x} ^ 2\right)}^{2} + 5 {\left({x}^{2} + \frac{1}{x}\right)}^{4} \left(2 + 2 + \frac{2}{x} ^ 3\right)$

#### Explanation:

We have
$f \left(x\right) = {\left({x}^{2} + \frac{1}{x}\right)}^{5}$
after the chain and the power rule we get
$f ' \left(x\right) = 5 {\left({x}^{2} + \frac{1}{x}\right)}^{4} \left(2 x - \frac{1}{x} ^ 2\right)$
$f ' ' \left(x\right) = 20 {\left({x}^{2} + \frac{1}{x}\right)}^{3} {\left(2 x - \frac{1}{x} ^ 2\right)}^{2} + 5 {\left({x}^{2} + \frac{1}{x}\right)}^{4} \left(2 + \frac{2}{x} ^ 3\right)$