Q:  Let f(x) = -2/x .  Find the equation of the tangent line when x = -1.

A:  First, let’s take the derivative….

f(x) = -2/x

Re-write this as:

f(x) = -2x^-1

Now, take the derivative using the power rule:

f ‘(x) = (-1)-2x^-1-1

f ‘(x) = 2x^-2

Now, plug in x = -1 to the derivative:

f ‘(-1) = 2(-1)^-2

f ‘(-1) = 2

So, the slope of the graph when x = -1 is 2….

To find the equation of the tangent line, I always like to start with:

y = mx + b

Our slope (m) is 2.. we just found that.  So,

y = 2x + b

Now we need to solve for “b” and we are done!  In order to do this, I need an (x, y) point.  Well, I know x = -1… so I need to find the y that goes along with that x… To do this, I go back to the original equation:

f(x) = -2/x

Plug in x = -1:

f(-1) = -2/-1 = 2

Therefore, my (x, y) point is (-1, 2)….

So,

y = 2x + b

Plug in my point:

2 = 2(-1) + b

2 = -2 + b

4 = b

Therefore, the equation of my tangent line when x = 1 is:

y = 2x + 4