Name: Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy
Uploaded: 2022-07-12T12:05:35.000Z
Duration: 15 min 43 s
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Adverbs of manner

We're first exposed to the idea of a slope of a line early on

in our algebra careers, but I figure it never hurts

Let me draw my x-axis, just like that, that is my x-axis.

And let me draw a line, let me draw a line like this.

And what we want to do is remind ourselves, how do we

And what we do is, we take two points on the line,

so let's say we take this point, right here.

Let's say that that is the point x is equal to a.

This would be the point f of a, where the function is

We could write f of x is going to be equal to mx plus b.

We don't know what m and b are, but this is all

And then the y-value is what happens to the function when

you evaluated it at a, so that's that point right there.

And then we could take another point on this line.

And then this coordinate up here is going to be

Because this is just the point when you evaluate

You put b in here, you're going to get that point right there.

So let me just draw a little line right there.

Actually, let me make it clear that this coordinate right

So how do we find the slope between these 2 points, or more

Because whole the slope is consistent the

And we know that once we find the slope, that's

actually going to be the value of this m.

That's all a review of your algebra, but how do we do it?

Well, a couple of ways to think about it.

You might have seen that when you first learned algebra.

Or another way of writing it, it's change in

So let's figure out what the change in why over the change

Well, let's just take, you can take this guy as being the

first point, or that guy as being the first point.

But since this guy has a larger x and a larger y,

The change in y between that guy and that guy is this

That distance right there is a change in y.

Or I could just transfer it to the y-axis.

Now what is your change in x The slope is change

Remember, we're taking this to be the first point,

so we took its y minus the other point's y.

So to be consistent, we're going to have to take this

And just like that, if you knew the equation of this line, or

if you had the coordinates of these 2 points, you would just

plug them in right here and you would get your slope.

And that comes straight out of your Algebra 1 class.

And let me just, just to make sure it's concrete for you, if

this was the point 2, 3, and let's say that this, up here,

was the point 5, 7, then if we wanted to find the slope of

this line, we would do 7 minus 3, that would be our change in

y, this would be 7 and this would be 3, and then we

Because this would be a 5, and this would be a 2, and so this

So 7 minus 3 is 4, and 5 minus 2 is 3. so your

And this is what the new concept that we're going

to be learning as we delve into calculus.

Let's see if we can generalize this somehow to a curve.

We have to have a curve before we can generalize

Well, actually, I want to leave this up here, show

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Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy

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