# What is the equation of the line tangent to  f(x)=2/(4 − x) at  x=p3?

Jan 20, 2018

$y = 2 x - 4$

#### Explanation:

Start by finding the derivative:

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

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

Now find $f ' \left(3\right)$ to get the gradient of the tangent:

$f ' \left(3\right) = \frac{2}{4 - 3} ^ 2 = \frac{2}{1} ^ 2 = 2$

Also find $f \left(3\right)$ to get a point of intersection:

$f \left(3\right) = \frac{2}{4 - 3} = \frac{2}{1} = 2$

So we have a line with gradient $m = 2$ and passes through the point: $\left(3 , 2\right)$. Now use $y - b = m \left(x - a\right)$

$y - 2 = 2 \left(x - 3\right) \to y = 2 x - 6 + 2$

$y = 2 x - 4$

I am assuming that the question actually says $x = 3$