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

Normal Line
$y = x$

#### Explanation:

Given $f \left(x\right) = 1 - \ln \frac{x}{x}$ and $x = 1$
find $f \left(1\right)$

$f \left(1\right) = 1 - \ln \frac{1}{1}$
$f \left(1\right) = 1$
Use point $\left(1 , 1\right)$

Find the slope of the Normal Line at $\left(1 , 1\right)$
slope =-1/f' (1)#

$f ' \left(x\right) = 0 + \left(- 1\right) \cdot \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x} ^ 2$
$f ' \left(1\right) = \left(- 1\right) \cdot \frac{1 - \ln 1}{1} ^ 2$
$f ' \left(1\right) = - 1$

slope m $= - \frac{1}{-} 1 = 1$

Use point slope form

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$y - 1 = 1 \left(x - 1\right)$

$y = x$