So Line L is tangent. Tangent to the circle.

So if I were to draw a tangent line right there, this line is normal to that tangent line.

line tangent to this point.

So if I were to draw the tangent line here the tangent

The line tangent to a curve

So this becomes a tangent line.

Construct a circle tangent to this line

Because this line is tangent to it.

The tangent line will look like this.

What's the equation of the tangent line?

I'm trying to draw a tangent line.

So that's my tangent line right there.

And the line, it really isn't a tangent.

But what is the slope of that tangent line?

The tangent line is something like that right there.

So what is the equation of the tangent line?

So this is the equation of the tangent line.

We have figured out the equation of the tangent line.

It means the slope of the tangent line is 0.

Select the line that the new circle should be tangent to...

And we know, that this line is tangent at point A.

The point where we're trying to find the tangent line, right?

Your slope over here is the slope of the tangent line.

And then we'll be finding the slope of the tangent line.

So over here, the tangent line has a pretty high let me do it in a different color over here, the tangent line has a pretty high positive slope.

So if I were to take that point and draw a tangent line a point that just grazes our curve if I were to just draw a tangent line.

A circle constructed by its center and tangent to a given line

When I say the slope you can imagine a tangent line here.

Then we're getting pretty close to the slope of the tangent line.

This is the slope at any particular x0 of the tangent line.

Or the slope of the tangent line at that point of the curve.

And by definition, the tangent line is going to go through that point.

A horizontal line would be tangent to the curve at y equals 0.

In the figure below, line AB is tangent to circle O at point A.

We have shown that, and visually, it shows that this is the tangent line.

So what's the slope of the tangent line at any point on this curve?

But I'm going to call this thing, which equals the slope of the tangent line, and it does equal the slope of the tangent line, I'm going to call this the derivative of f.

So if we find a line that's tangent maybe there and there on the hyperbola, then we might have found our common tangent with positive slope.

All it really is derivatives is the slope of the tangent line at any point.

I could use the derivative function to figure out the slope of the tangent line.

We want to know equation of the tangent line at x is equal to 1.

Well, this is the slope of the tangent line at x is equal to 7.

The tangent line is this thing right here that I want to find the slope of.

So this point, 1 comma e, this is where we want to find the tangent line.

And then, as h approaches 0, this secant line is going to get closer and closer to the slope of the tangent line.