Answer
$y'= 5(x+\frac{1}{x})^4(1-\frac{1}{x^2})$
Work Step by Step
Derive the expression using the chain rule:
$y=(x+\frac{1}{x})^5$
Apply the chain rule: $f(g(x)) = f'(g(x))g'(x)$:
Set $g(x)$ equal to $u$
$g(x) = u = x + \frac{1}{x}$
$g'(x) = u' = 1 - \frac{1}{x^2}$
$y(u) = (u)^5 u'$
$y'(u) = 5(u)^4u'$
$y' = 5(x + \frac{1}{x})^4 ( 1 - \frac{1}{x^2})$
