The tangent slope.

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

The tangent line will look like this.

line tangent to this point.

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.

Sohcahtoa tangent deals with opposite over adjacent.

So my slope of my tangent of my secant

And it would tell me the slope of the tangent

So what is the equation of the tangent line?

We have figured out the equation of the tangent line.

What's the equation of the tangent line?

So this becomes a tangent line.

So that's my tangent line right there.

So this is the equation of the tangent line.

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

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

The tangent line is something like that right there.

So the equation of the tangent line, its slope is going to be the derivative at this point.

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

It means the slope of the tangent line is 0.

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

So it says, find the equation of the tangent line to the function f of x is equal to x e to the x at x is equal to 1.

It means that the slope of the tangent line at x is equal to 1, to this curve, to the function, is minus 2.

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

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

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.

Akosh ? sent me a problem where it actually wants you to find the equation of the tangent line.

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

So the first thing we want to know is the slope of the tangent line, and that's just the derivative at this point.

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

I've told you multiple times that the derivative of a curve at a point is the slope of the tangent line, but our friend ?

Let's use this little graphing calculator to confirm that that really is the equation of the tangent line.

And if we want to get the slope of the tangent line, we would just have to find what happens as h gets smaller and smaller and smaller.

So this will be a little bit over 1, this looks like that will be the slope of the tangent line right over there.

When you're tangent at point A, that means your tangent to the radius at that point.

Tangent means that it just touches the outside of the circle right there at only one point.

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

I'm not being very accurate or precise right there, but right about there, you see that the slope of the tangent line

Really, we just want to, if we want to find the slope of this tangent line, we just have to find the limit of this value as h approaches 0.

Or if we went all the way over to 2.25, the slope will be aproximatly 30 right over there, so once you have the curve you now have the slope of the tangent line or if you have the slope of the curve at every point we see over here.

And then we use what we learned from algebra 1 to figure out its equation, and we'll graph it here, just to confirm that we actually figured out the equation of the tangent line.

It says, ok you have done it right and it then drew the whole slope of the derivative and what the entire derivative is saying, is at any point this is the slope of the tangent line.

So what we need to do is move this so it becomes pretty close, or as close as we can eye ball it to the slope of the tangent line. and so that looks about right, as it looks about right.

So at that point, the slope of the tangent line, or the slope of the actual graph, or the instantaneous slope, however you want to view it let me see how well I can draw it it'll look something like that, and that's the tangent line.