# What is the equation of the normal line of f(x)=x^2-x+1/x at x = 2?

Dec 2, 2016

The equation is:

$y \left(x\right) = - \frac{4}{11} x + \frac{71}{22}$

#### Explanation:

The general formula of the normal line to a curve is:

$y \left(x\right) = f \left(\overline{x}\right) - \frac{1}{f ' \left(\overline{x}\right)} \left(x - \overline{x}\right)$

Given:

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

For $x = 2$:

$f \left(x\right) {|}_{x = 2} = \frac{5}{2}$

$f ' \left(x\right) {|}_{x = 2} = \frac{11}{4}$

So:

$y \left(x\right) = \frac{5}{2} - \frac{4}{11} \left(x - 2\right) = - \frac{4}{11} x + \frac{71}{22}$