# What is the equation of the normal line of f(x)=2x+3/x at x=1?

##### 1 Answer
Oct 31, 2016

$\therefore y = x + 4$

#### Explanation:

$f \left(x\right) = 2 x + \frac{3}{x}$
$\therefore f \left(x\right) = 2 x + 3 {x}^{-} 1$

Differentiating wrt $x$ gives us:
$f ' \left(x\right) = 2 + 3 \left(- {x}^{-} 2\right)$
$\therefore f ' \left(x\right) = 2 - \frac{3}{x} ^ 2$

When $x = 1 \implies f \left(1\right) = 2 + 3 = 5$
and, $f ' \left(1\right) = 2 - 3 = - 1$

So the gradient of the tangent at $x = 1$ is $m ' = - 1$, and the tangent and normal are perpendicular so the product of their gradients is $- 1$
Hence, gradient of Normal is $m = - \frac{1}{- 1} = 1$

So the Normal has gradient $m = 1$ and it passes through $\left(1 , 5\right)$, so using $y - {y}_{1} = m \left(x - {x}_{1}\right)$ the Normal equation is given by:

$y - 5 = \left(1\right) \left(x - 1\right)$
$\therefore y = x + 4$