# How do you find the equation of a line tangent to y=4/x at (2,2)?

Aug 24, 2016

We know that the tangent passes through the point $\left(2 , 2\right)$, so we only need to find out the slope of the tangent at that point.

#### Explanation:

But the slope of the tangent of $y$ at a given point is the value of the first derivative at that point. Now, for this function:

$y ' = - \frac{4}{x} ^ 2$, and so evaluating at $x = 2$ we get

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

Then we now know that the tangent is $y = \left(- 1\right) \cdot x + b$, and since the point $\left(2 , 2\right)$ is on the line we also know that:

$2 = \left(- 1\right) \cdot 2 + b$. Then $b = 4$ and the equation of the tangent is:

$y = - x + 4$