# What is the equation of the normal line of f(x)= -2x^3-10x  at x=2 ?

Jan 12, 2017

The equation of the normal line is:

$y = \frac{x}{34} - 36 - \frac{1}{17}$

#### Explanation:

The equation of the line normal to the graph $y = f \left(x\right)$ in the point $\left(\overline{x} , f \left(\overline{x}\right)\right)$ is given by:

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

In our case $\overline{x} = 2$ and:

$f \left(x\right) = - 2 {x}^{3} - 10 x \implies f \left(2\right) = - 36$

$f ' \left(x\right) = - 6 {x}^{2} - 10 \implies f ' \left(2\right) = - 34$

So the equation of the normal line is:

$y = - 36 + \frac{1}{34} \left(x - 2\right) = \frac{x}{34} - 36 - \frac{1}{17}$