Name: Calculating slope of tangent line using derivative definition | Differential Calculus | Khan Academy
Uploaded: 2022-07-12T12:05:43.000Z
Duration: 8 min 28 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

In the last video we tried to figure out the slope of a

point or the slope of a curve at a certain point.

And the way we did, we said OK, well let's find the slope

between that point and then another point that's not too

And it looks all fancy, but this is just the y value of the

point that's not too far away, and this is just the y value

point of the point in question, so this is

And then you divide that by your change in x.

So in the example we did, h was the difference

We said hey, what if we take the limit as this point

right here gets closer and closer to this point.

If this point essentially almost becomes this point, then

our slope is going to be the slope of our tangent line.

And we define that as the derivative of our function.

So let's if we can apply this in this video to maybe make

things a little bit more concrete in your head.

First I'll do a particular case where I want to find the

Let's say I have the curve-- this is the curve-- y

So this is my y-axis, this is my x-axis, and I want to know

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

You can imagine a tangent line that goes just like that, and

it would just barely graze the curve at that point.

But what is the slope of that tangent line?

What is the slope of that tangent line which is the same

as the slope of the curve right at that point.

So to do it, I'm actually going to do this exact technique that

we did before, then we'll generalize it so you don't have

to do it every time for a particular number.

I'm changing the notation because in some books you'll

see an h, some books you'll see a delta x, doesn't hurt to

So first of all what is this point right here?

This is a curve y is equal to x squared, so f of x is 3

So if we were go all the way up here, what is that point?

It's the same thing as this one right here,

I could have called this 3 plus h just as easily.

Well whatever x value is, it's on the curve, it's going

So it's going to be the point 3 plus delta x squared.

So let's figure out the slope of this secant

And let me zoom in a little bit, because that might help.

So if I zoom in on just this part of the curve,

And then I have one point here, and then I have the

This was the point over here, the point 3,9.

And then this point up here is the point 3 plus delta x, so

just some larger number than 3, and then it's going to

So it's going to be 3 plus delta x squared.

I'm just foiling this out, or you do the distribute

a plus b squared is a squared plus 2 a b plus b squared, so

it's going to be 9 plus two times the product

So plus 6 delta x, and then plus delta x squared.

That's the coordinate of the second line.

This looks complicated, but I just took this x value and I

squared it, because it's on the line y is equal to x squared.

line is going to be the change in y divided

So the change in y is just going to be this guy's y value,

which is 9 plus 6 delta x plus delta x squared.

That's this guy's y value, minus this guy's y value.

And you want to divide that by your change in x.

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Calculating slope of tangent line using derivative definition | Differential Calculus | Khan Academy

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