# What is the equation of the normal line of f(x)=1/x^2-x at x = 4?

Nov 15, 2015

$y = \frac{32}{33} x - \frac{4127}{528} \Leftrightarrow 528 y = 512 x - 4127$

#### Explanation:

$f \left(4\right) = \frac{1}{16} - 4 R i g h t a r r o w P = \left(4 , - \frac{63}{16}\right)$

$\frac{d}{\mathrm{dx}} f \left(x\right) = - 2 {x}^{-} 3 - 1$

$\frac{d}{\mathrm{dx}} f \left(4\right) = - \frac{2}{4} ^ 3 - 1 = \frac{- 1 - 32}{32} = m$ // inclination of the tangent line

$a = - \frac{1}{m} = \frac{32}{33}$ // inclination of the normal line $n$

$P \in n : y = a x + b$ // let's determine b

$- \frac{63}{16} = \frac{32}{33} \cdot 4 + b$

$b = - \frac{63}{16} - \frac{128}{33} = \frac{- 63 \cdot 33 - 128 \cdot 16}{16 \cdot 33} = - \frac{4127}{528}$