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

Mar 19, 2017

$y = - \frac{4}{7} x - \frac{79}{14}$

#### Explanation:

slope function $= f ' \left(x\right)$

Find the derivative:
Use the Quotient Rule $\left(\frac{u}{v}\right) ' = \frac{v u ' - u v '}{v} ^ 2$

Let $u = 2 {x}^{2} + 1$, $u ' = 4 x$
Let $v = x$, $v ' = 1$

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

Find the tangent and normal slopes:
Tangent Slope, $m = f ' \left(- 2\right) = \frac{2 {\left(- 2\right)}^{2} - 1}{- 2} ^ 2 = \frac{7}{4}$

Normal slope, $- \frac{1}{m} = - \frac{4}{7}$

Find normal equation:
1. Find the point: $f \left(- 2\right) = \frac{2 {\left(- 2\right)}^{2} + 1}{-} 2 = - \frac{9}{2}$
2. Use $y - {y}_{1} = m \left(x - {x}_{1}\right)$: $y - \left(- \frac{9}{2}\right) = - \frac{4}{7} \left(x - \left(- 2\right)\right)$
3. Simplify: $y + \frac{9}{2} = - \frac{4}{7} x - \frac{8}{7}$;
4. write in $y$-intercept form: $y = - \frac{4}{7} x - \frac{79}{14}$