# What is the equation of the line normal to  f(x)=lnx-1/x at  x=3?

Jan 8, 2016

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

now the line equation is:

$\left(y - {y}_{0}\right) = m \left(x - {x}_{0}\right)$

where:

${x}_{0} = 3$
${y}_{0} = f \left({x}_{0}\right)$
$m = - \frac{1}{f ' \left({x}_{0}\right)}$ because we looking for the normal line

then:

$f ' \left(3\right) = \frac{3 - 1}{9} = \frac{2}{9}$
$- \frac{1}{f ' \left(x\right)} = - \frac{9}{2}$
$f \left({x}_{0}\right) = \ln 3 - \frac{1}{3}$

$y - \left(\ln 3 - \frac{1}{3}\right) = - \frac{9}{2} \left(x - 3\right)$
$y = - \frac{9}{2} x - \frac{27}{2} + \left(\ln 3 - \frac{1}{3}\right)$