The Tangent Line

As you take smaller and smaller distances, of course, this line slightly changes.
But what happens when this distance becomes zero?

Of course, just substituting for the slope won't work. But, this is why we learned limits.
You will instead play around with the limit of this ratio, as this distance gets infinitesimally small.

So let's try to do that again for our previous
Let's start from, say, x = 2.

Your starting point is of course, , so

Now let's suppose you're travelling some kind of distance, to whatever other point. Let's name this distance some variable .

As you travel, your slope of whatever line, will be:
You've probably already figured it out, but let's keep going.

We said that this distance will be getting infinitesimally small, so, we want the slope of the tangent line.
It's a tangent line because at a distance of zero, there will be only one point. So this tangent line only has this exact point in common with the graph.

We chose x = 2. Let's see where that infinitesimally small distance gets us.

So, when taking a very small distance, we go from a line on two points, to the rate of change at a point.
So, at any given point, we can attempt to calculate this.
The slope of the tangent line does not represent the average rate of change anymore, because it wouldn't make sense.
It now represents the instantaneous(at a single point, at a given moment) rate of change.