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Let's now expand our knowledge of calculus

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to the third dimension.

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So first of all, just what does a function look like

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in three dimensions?

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And actually we'll go over the different types.

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Because you can have a line in three dimensions, or kind of

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a curve in three dimensions.

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You can have a surface.

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You could have a vector field.

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There are different types of representations we'll see,

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when we start working with three dimensions.

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But I think the most intuitive-- and none of these

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are directly intuitive-- I think you have to really be

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able to visualize them.

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But the most intuitive, to me at least, is a surface

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in three dimensions.

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And eventually, we can expand this into n dimensions.

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But then it becomes very hard to visualize.

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So we had our traditional x and y-axis before, but

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now let's give another dimension of height.

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Let's say that this is my x-axis-- and I'll draw

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the positive quadrant.

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That's my x-axis.

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That's the y-axis, and that's the z-axis.

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And the convention is to kind of follow the right hand rule,

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where the x-axis-- taking the cross product of the x-axis

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with the y-axis-- is equal to the z-axis.

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What do I mean by that?

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This is x-- those colors really don't go well together.

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This is y.

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This is z.

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What do I mean by the cross product?

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So if this is the unit vector in the x

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direction-- so that's i.

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Let's say this is a length of 1.

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This is in the y-direction, so it's j.

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Oops.

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j, little cap.

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And that cap just means it's a unit vector.

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It's in y-direction, but it has a magnitude of 1.

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And I'll use a different color for z.

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z is up.

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And the unit vector for there is k.

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And this is just a convention, that i cross j is equal to k.

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And that's just the convention, you know,

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for drawing the x-axis.

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Do we make increasing x pointing out this way?

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Or do we make increasing x point inwards?

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And this gives us the convention.

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So i goes in this direction.

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I'm trying to make sure I can do my hand properly.

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So let me draw the cross product.

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So if you take the first vector, put your index finger

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in the direction of the first vector, middle finger in the

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direction of the second vector, and your other fingers can

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do what they need to do.

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So this is going in the direction of i.

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That is going in the direction of k.

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Sorry, of j, right?

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In the y-direction.

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And then you have your palm of your thumb, and then your thumb

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is going to open up in this direction.

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Your thumb is going to point up, which is

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the direction of k.

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So that's just a-- good to know, where did that

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convention come from?

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This is kind of called a right-handed coordinate system.

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But let's get to the meat and potatoes.

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So how do we define a surface in three dimensions?

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Well, we can define z as a function of x and y.

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So let's do that.

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And just the notation, z is equal to a function of x and y.

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And all that means is that if I give you an x-value and a

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y-value you get a z-value.

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And so, I don't know, let's pick one.

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z is equal to x plus y.

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So this z is equal to, I don't know, x squared plus y.

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So how do we plot the points on the surface?

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And I'll show you, actually, computer generated surfaces

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that are far more professional looking than anything I

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could possibly try to draw.

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So let's say, what's f of-- f of, I don't know, 2 comma 1.

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Well, that would mean x is 2, so it's 2 squared plus 1.

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Well, it equals 5.

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And if we had to plot that point of the surface-- and

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maybe I'll actually graph this one in a little bit-- we

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go along the x-axis 2.

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So 1, 2.

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We go along the y-axis 1.

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So if you take this point-- this is x is equal

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to y is equal to 1.

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And then we go and then we say, well, z is equal to 5.

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So we can go up here, I don't know, we'd go up 5 units.

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And we would plot that point.

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And you would see, if you kept doing that, you

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would plot a surface.

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You'd plot a surface.

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And let me clean this up a little bit.

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So the natural question that you might want to ask--

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and actually, let me show you a surface.

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I'm afraid that when I manipulate this graph, it'll

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slow down my computer and I'll start sounding

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like I'm melting.

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But I'll take that risk.

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Just bear with me.

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So here is a surface I use, using this Java applet grapher.

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And it's actually free.

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I'll give you the link for it.

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But this surface right here, this is-- I'll actually show

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you the graph of this.

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It'll start taking the partial derivative.

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Don't worry about this wall.

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We'll get to this in a second.

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But this is a function of x and y.

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You can see this is the x-axis, the y-axis.

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The height is the z-axis.

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This'll probably really slow down my computer, but you

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can actually rotate it.

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Look at that.

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I don't want to slow.

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I don't want to slow things too down while I'm trying

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to do my screen capture.

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Anyway, I think you'd understand, where you pick an

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x-point, you pick a y-point, and then z-- this surface right

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here, without this line intersecting it-- this

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surface right here, is a function of x and y.

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So the question is, well, how do we apply

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calculus to surfaces?

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Because-- actually, let me bring that thing out again.

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Because if you look at this surface, if you were to pick

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any arbitrary point on this surface, and say, what is

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the slope of that surface?

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Well, it kind of has no meaning, because you have to

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kind of pick a direction.

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If you said, what is the slope of the tangent line?

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Any point on this graph actually has an infinite

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number of tangent lines.

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I mean, think of it this way.

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Take a bowl or something that maybe-- you know, like this.

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And then take a, I don't know, a toothpick.

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And make that toothpick tangent to the bowl, and you can see

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that on any point on the bowl, you can just rotate

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that toothpick around.

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So you kind of have to pick the orientation of that toothpick.

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So what we're going to learn is when you take a derivative in

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three dimensions, you have to specify the direction that

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you're taking the derivative in.

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And this is why I actually drew this wall here.

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This wall is the equation y is equal to 0.3.

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So you can kind of view it.

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Along this wall, y is a constant, right?

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So if we assume that y is constant, then maybe we could

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take just the derivative with respect to x.

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So we would essentially take the slope of this

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curve right here.

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And let's figure out how to do it.

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So first of all, what is the equation of this surface?

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And I just picked one that they had on Wikipedia.

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But the equation of that surface is-- and I'm going to

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remove this now, so I don't sound like I'm melting.

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The equation of that surface-- and let me just clear out

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everything, just because we'll probably need the extra space.

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Go back to the pen tool.

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The equation is z is equal to x squared plus xy plus y squared.

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So we said if we want to take a derivative, it's hard to-- you

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know, you can't just say there is one derivative.

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We have to pick a direction.

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We have to hold everything else constant and take the

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derivative with respect to just one variable.

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And that is called the partial derivative.

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I know it sounds fancy, but you'll see.

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It's actually no harder than taking a regular derivative.

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You just have to make sure you remember which variable is a

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variable, and which one is a constant.

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So let's say we wanted to hold y constant.

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And we just say, for any constant y, how much does z

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change with respect to x?

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Then we take the partial derivative-- this

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is the notation.

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You can view it as a d with the top curled.

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The partial derivative of z with respect to x.

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It equals-- all we do is we take this expression-- we take

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the derivative of x-- and we just assume that y

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is some constant.

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So what's the derivative of 2x with respect to x?

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Well, it's just 2x.

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What's the derivative of xy with respect to x?

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Well, y is just a number.

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It's just a constant.

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Remember, we're taking an implicit derivative here.

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y is just a constant.

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So if you have some constant times x, the derivative of

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that is just the constant.

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Plus y.

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And then what's the derivative of y squared with respect to x?

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Well, we're assuming y squared is a constant.

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It's just a number, right? y is just a number.

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So the derivative of just the number with respect

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to x is just 0.

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So the derivative of that is 0.

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So the partial derivative of z with respect to x is 2x plus y.

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Now, what does that mean?

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Well, that means if I were-- and actually, let me give you

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a little notation, before I show you what that means.

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Another way to write this exact same thing is if we wrote that

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f of xy is equal to the same thing-- x squared plus xy plus

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y squared-- the partial of f with respect to x we could

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have written as this.

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The partial derivative of f with respect to x-- and still

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a function of x and y, right?

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It still depends on what constant y you're using--

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is equal to 2x plus y.

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Anyway, I thought it's nice to see that notation.

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Now, what does this mean?

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Well, what is the slope of z with respect to x at, say,

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when x is 1 and-- actually, let's pick smaller numbers.

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When x is equal to, I don't know, when x is equal to 0.2

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and y is equal to, I don't know, 0.3.

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Well, we could use this.

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The partial derivative of f, with respect to x, at the

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point 0.2, 0.3 is equal to 2 times x-- that's 0.4--

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plus y-- plus 0.3.

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So the slope of this function with respect to x at the 0.2,

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comma 0.3, is equal to 0.7.

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Let's see if we can visualize that.

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So that wall represents the line y is equal to 0.3.

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And we want the slope at equal at-- x is equal to 0.2.

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So this is x is 0.2, right here.

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So the rate at which the height, or the rate at

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which z is changing with respect to x, is 0.7.

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So every time x increases 1, z will increase by 0.7.

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So the slope is a little bit less than 1.

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I think you see that, right?

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The tangent right here is increasing with increasing

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values of x, but a little bit less than 45 degrees.

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Anyway, I'm all out of time.

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See you in the next video.

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