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In the last video, we found the slope at a particular point of

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the curve y is equal to x squared.

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But let's see if we can generalize this and come up

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with a formula that finds us the slope at any point of the

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curve y is equal to x squared.

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So let me redraw my function here.

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It never hurts to have a nice drawing.

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So that is my y-axis.

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That is my x-axis right there.

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My x-axis.

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Let me draw my curve.

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It looks something like that.

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You've seen that multiple times.

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This is y is equal to x squared.

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So let's be very general right now.

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Remember, if we want to find-- let me just write the

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definition of our derivative.

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So if we have some point right here-- let's call that x.

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So we want to be very general.

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We want to find the slope at the point x.

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We want to find a function where you give me an x

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and I'll tell you the slope at that point.

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We're going to call that f prime of x.

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That's going to be the derivative of f of x.

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But all it does is, look, f of x, you give-- it's a function

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that you give it an x, and it tells you the value of that.

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And we draw the curve here.

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With f of x, you give that same x but it's not going to tell

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you the value of the curve.

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It's not going to say, oh, this is your f of x.

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It's going to give you the value of the slope of

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the curve at that point.

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So f of x, if you put it into that function, it should tell

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you, oh, the slope at that point, is equal to-- you know,

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if you put 3 there, you'll say, oh, the slope there

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is equal to 6.

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We saw that in the last example.

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So that's what we want to do.

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And we saw on the last-- I think it was 2-- videos ago,

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that we defined f prime of x to be equal to-- just the--

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well, I'll write it this way.

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It's the slope of the secant line between x and some

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point that's a little bit further away from x.

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So the slope of the secant line is change in y.

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So it's the y value of the point that's a little

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bit further away from x.

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So f of x plus h minus the y value at x, right?

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Because this is right here.

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This is f of x.

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So minus f of x.

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All of that over the change in x.

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So if this is x plus h here, the change in x

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is x plus h minus x.

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Or this distance right here is just h.

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The change in x is going to be equal to h.

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So that's just slope of the secant line, between

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any 2 points like that.

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And we said, hey, we could find the slope of the tangent line

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if we just take the limit of this as it approaches--

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as h approaches 0.

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

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Now let's apply this idea to a particular function, f of

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x is equal to x squared.

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Or y is equal to x squared.

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So here, we could have the point-- we could consider this

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to be the point x-- x squared.

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So f of x is just equal to x squared.

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And then this would be the point-- let me do it in

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a more vibrant color.

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This is the point x plus h-- that's this point right here.

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It's a little bit further down.

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And then x plus h squared.

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And you know, in the last video, we did this

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for a particular x.

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We did it for 3.

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But now I want a general formula.

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You give me any x and I won't have to do what I did in the

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last video for any particular number.

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I'll have a general function.

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You give me 7, I'll tell you what the slope is at 7.

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You give me negative 3, I'll tell you what the slope

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is at negative 3.

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You give me 100,000, I'll tell you what the

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slope is at 100,000.

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So let's apply it here.

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So we want to find the change in y over the change in x.

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So first of all, the change in y is this guy's y value,

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which is x plus h squared.

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That's this guy's y value right here.

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That's this right here.

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That's x plus h squared.

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I just took x plus h, evaluated, I squared it, and

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that's its point on the curve.

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So it's x plus h squared.

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So that's there right there.

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And then what's this value?

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f of x right here is equal to-- I know it's getting messy--

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is equal to x squared.

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If you take your x, you evaluate the function

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at that point, you're going to get x squared.

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So it's equal to minus x squared.

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This is your change in y.

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That's this distance right there.

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And just to relate it to our definition of a derivative,

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this blue thing right here is equivalent to this

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thing right here.

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We just evaluated our function.

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Our function is f of x is equal to x squared.

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We just evaluated when x is equal to x plus h.

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So if you have to square it, if I put an a there,

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it'd be a squared.

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If I put an apple there, it'd be apple squared.

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If I put an x plus h in there, it's going to

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be x plus h squared.

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So this is that thing.

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And then, this thing right here is just the function evaluated

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at the point in question.

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Right there.

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So this is our change in y.

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And let's divide that by our change in x.

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Our change in x-- if this is x plus h and this is just x, our

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change in x is just going to be h.

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So that's where we get that term from.

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So this is just a slope between these 2 points.

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This is just a slope between those two points.

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But, of course, we want to find-- the limit at this point

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gets closer and closer to this point, and this point gets

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closer and closer to that point.

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So this becomes a tangent line.

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So we're going to take the limit as h approaches 0, and

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this is our f prime of x.

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And this is the exact same definition of this, instead

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of being general and saying, for any function, we know

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what the function was.

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It was f of x is equal to x squared.

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So we actually applied it.

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Instead of f of x, we wrote x squared.

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Instead of f of x plus h, we wrote x plus h squared.

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So let's see if we can evaluate this limit.

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So this is going to be equal to the limit as h approaches

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0 to square this out.

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I'll do it in the same color.

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That's x squared plus 2xh plus h squared, and then we have

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this minus x squared over here.

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I just multiplied this guy out over here.

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And then all of that is divided by h.

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Now let's see if we can simplify this a little bit.

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Well, you immediately see you have an x squared and you

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have a minus x squared, so those cancel out.

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And then we can divide the numerator and the

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denominator by h.

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So this simplifies to-- so we get f prime of x is equal to--

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if we divide the numerator and the denominator by h--

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we get 2x plus h.

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I'm sorry, I forgot my limit.

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It equals the limit.

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Very important.

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Limit as h approaches 0 of divide everything by h, and

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you get 2x plus h squared divided by h is h.

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And if you remember the last video, when we did it with a

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particular x, we said x is equal to 3, we got 6

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plus delta x here.

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Or 6 plus h here, so it's very similar.

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So if you take the limited h approaches 0 here, that's

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just going to disappear.

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So this is just going to be equal to 2x.

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So we just figured out that if f of x-- this is a big result.

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This is exciting!

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That if f of x is equal to x squared, f prime

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of x is equal to 2x.

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That's what we just figured out.

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And I wanted to make sure you understand

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how to interpret this.

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f of x, if you give me a value, is going to tell you the value

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of the function at that point.

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At prime of x it's going to tell you the

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slope at that point.

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Let me draw that.

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Because this is a key realization.

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And you might, you know, it's kind of maybe initially

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unintuitive to think of a function that gives us the

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slope, at any point, of another function.

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So it looks like this.

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Let me draw a little neater than that.

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Ah, it's still not that neat.

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That's satisfactory.

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Let me just draw it in the positive coordinate.

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Well, I'll just draw the whole-- the curve looks

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something like that.

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Now this is the curve of f of x.

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This is the curve of f of x is equal to x squared.

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Just like that.

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So if you give me a point.

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You give me the point 7.

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You apply, you put it in here, you square it.

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And it is mapped to the number 49.

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So you get the number 49 right there.

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This is the number 7, 49.

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You're used to dealing with functions right there.

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But what is f prime of 7?

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f prime of 7.

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You say, 2 times 7 is equal to 14.

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What is this 14 number here?

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What is this thing?

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Well, this is the slope of the tangent line

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at x is equal to 7.

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So if I were to take that point and draw a tangent line-- a

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point that just grazes our curve-- if I were to just

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draw a tangent line.

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That wasn't tangent enough for me.

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So that's my tangent line right there.

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You get the idea.

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The slope of this guy-- you do your change in y over your

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change in x-- is going to be equal to 14.

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The slope of the curve at y is equal to 7-- is

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a pretty steep curve.

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If you wanted to find the slope, let's say that this

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is y-- let's say it's x is equal to 2.

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I said at x is equal to 7, the slope is 14.

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At x is equal to 2, what is the slope?

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Well, you figure out f prime of 2, which is equal to 2 times

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2, which is equal to 4.

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So the slope here is 4.

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You could say m is equal to 4. m for slope.

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What is f prime of 0?

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f prime.

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We know that f of 0 is 0, right?

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0 squared is 0.

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But what is f prime of 0?

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Well, 2 times 0 is 0.

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That's also equal to 0.

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But what does that mean?

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What's the interpretation?

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It means the slope of the tangent line is 0.

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So a 0 sloped line looks like this.

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Looks just like a horizontal line.

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And that looks about right.

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A horizontal line would be tangent to the

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curve at y equals 0.

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Let's try another one.

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Let's try the point minus 1.

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So let's say we're right there. x is equal to minus 1.

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So f of minus 1, you just square it.

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Because we're dealing with x squared.

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So it's equal to 1.

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That's that point right there.

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What is f prime of minus 1?

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f prime of minus 1 is 2 times minus 1.

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2 times minus is minus 2.

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What does that mean?

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It means that the slope of the tangent line at x is equal to

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1, to this curve, to the function, is minus 2.

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

258
00:10:54,600 --> 00:10:57,139
line looks like that-- and look, it is a downward

259
00:10:57,139 --> 00:10:57,699
sloping line.

260
00:10:57,700 --> 00:10:58,450
And it makes sense.

261
00:10:58,450 --> 00:11:04,700
The slope here is equal to minus 2.

262
00:11:04,700 --> 00:11:04,733