# What is the equation of the normal line of f(x)=5x^3+2x^2+x-2 at x=2?

Sep 18, 2016

$x + 69 y = 3314$

#### Explanation:

$f \left(2\right) = 40 + 8 + 2 - 2 = 48$

The tangent line is: $y = k x + n$, where $k = f ' \left(x\right)$. So,

$f ' \left(x\right) = 15 {x}^{2} + 4 x + 1$

$f ' \left(2\right) = 69 \implies k = 69$

The normal line is: $y = {k}_{1} x + {n}_{1}$, where ${k}_{1} = - \frac{1}{k}$.

${k}_{1} = - \frac{1}{69}$

$48 = - \frac{1}{69} \cdot 2 + {n}_{1} \implies {n}_{1} = 48 + \frac{2}{69} = \frac{3314}{69}$

Finally,

$y = - \frac{1}{69} x + \frac{3314}{69}$ or

$x + 69 y = 3314$