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Welcome back.

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I'm now going to do some more examples of a bit of a review

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of some of the derivatives that we've been seeing.

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And then I'll introduce you to something called the chain rule

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which expands the universe of the types of functions we can

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take the derivatives of.

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So in the last presentation, I showed you how to function if I

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had f of x is equal to 10x to the seventh plus 6x to the

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third plus 15x minus x to the 16th.

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To take the derivative of this entire function, we take just

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the derivatives of each of the pieces, right?

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Because you can add them up.

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So f prime of x in this example, is equal to-- and

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I think you get the hang of it at this point.

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It's actually fairly straightforward.

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We take the 7, multiply it by the 10.

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So we get 70x, then 1 degree less.

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So 70x to the sixth plus 18x squared plus 15.

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We can kind of view this as x to the 1, right?

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So it's 1 times 15 times x to the 0.

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Which is 1.

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So that's just 15 minus 16x to the 15th.

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And I don't want you to lose sight of what we're

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actually doing here.

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What is f prime of x?

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This is the function the tells us the slope of any point

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x, along the curve f of x.

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It's a pretty interesting thing.

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Let me just draw to maybe give you a little bit of intuition.

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I don't know what the slope of f of x really looks like.

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And actually, let's pretend like this isn't f of x.

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Let's pretend like this is just some arbitrary

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function I'm drawing.

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If this is f of x, just some curve that does all sorts of

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crazy things, f prime of x tells me the slope at any

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point along that line.

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So if I wanted to know the slope at this point right here,

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I could use the derivative function to figure out the

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slope of the tangent line.

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The tangent line is something like that right there.

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Or if I wanted to figure out the slope at this point,

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once again I'd use the derivative function.

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And it would tell me the slope of the tangent

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line at that point.

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Which would be something like that.

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So it's a pretty useful thing.

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And once I give you all the tools to analytically solve a

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whole host of derivatives, then we'll actually do a bunch

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of word problems and applications of derivatives.

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And I think you'll see that it's a really, really,

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really useful concept.

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So let's move on.

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I think you get the idea of how to do these derivatives

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of polynomials.

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Let me erase this.

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I'm actually using a different tool now.

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So I think it might be a bit easier.

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Let's see, someone was calling me.

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But you're more important so I will not answer the phone.

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I'm going to introduce you-- this tool doesn't have, I

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don't think it has a straight up eraser.

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Actually, maybe let's see.

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If I do it like this.

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Oh let me see.

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No that doesn't work.

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Let me just erase like this, the old-fashioned way.

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You just have to bear with me.

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And then once I finish erasing, I will show you the chain rule.

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This is good.

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It feels like I'm a real teacher with a real chalkboard

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and a real eraser now.

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This is a lot cleaner than a normal chalkboard as well.

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Bear with me, almost there.

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I'll figure out a faster way to do this over the

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next couple of videos.

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It's pretty sad.

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I'm showing you how to do derivatives in calculus, but

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I don't know how to erase a faster way than this.

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There, we're done.

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OK.

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So now I'm going show you how to solve the derivatives of a

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slightly more complicated type of a function.

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It's actually not more complicated.

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It's just different.

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So let's say f of x is equal to 2x plus 3 to the fifth power.

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And I want to figure out the derivative of this.

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We're going to use something called the chain rule.

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Because one thing we could do, we could just multiply out 2x

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plus 3 to the fifth power.

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And if you've ever done that, you know it's a pain.

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So that's not something we'd want to do.

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So we're going to use something called the chain rule.

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And I'm just going to give you a bunch of examples before I

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even show you the definition of the chain rule.

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Because I think this is something that you just

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have to learn by example.

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So the chain rule just tells us that the derivative of let's

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say this function right here.

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You take the derivative of the subfunctions, and then you can

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take a derivative of the entire function.

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I'll tell you that formally.

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But I think when you introduce it formally, it gets

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more confusing.

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So what I do, I just take the derivative of 2x plus 3 first.

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And actually let me use colors.

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I think that might simplify it.

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So I take the derivative of 2x plus 3.

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What's the derivative of 2x plus 3?

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Well you know that.

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It's just the derivative of 2x, which is 2.

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And then the derivative of 3 is 0.

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So the derivative of 2x plus 3 is just 2.

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And then I'm going to multiply that times the derivative

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of the whole thing.

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And I just pretend like 2x plus 3 is just like

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a variable by itself.

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So then what's the derivative of x to the fifth?

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Well the derivative of x to the fifth-- I'm going to do that in

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a different color-- the derivative of x to the

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fifth is 5x to the fourth.

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So it'll be 5 times something to the fourth.

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But here we didn't take the derivative of x the fifth.

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We took the derivative of 2x plus 3 to the fifth.

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So we just put the 2x plus 3 there instead.

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So what did we do here?

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We went in the inside of the function, and we took

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the derivative here.

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And the derivative of 2x plus 3 was just 2.

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And then we multiplied it by the derivative of

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the greater function.

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And we just pretended like the 2x plus 3 was a variable.

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It was like x.

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So instead of 5x to the fourth, we got 5 times

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2x plus 3 to the fourth.

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And if we just simplify that, f prime of x is equal to

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2 times 5 is 10; 10 times 2x plus 3 to the fourth.

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That was a lot simpler than multiplying out 2x plus 3 to

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the fifth power, and then doing the derivatives the old way.

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I know this was probably a little confusing to you,

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so I'm going to try to do a couple more examples.

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Let's say I had g of x is equal to x-squared plus 2x plus

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3 to the eighth power.

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So g prime of x is going to equal-- well what did we say?

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We take the derivative of the inside.

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This is called the chain rule.

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What's the derivative of the inside?

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It's 2x plus 2 plus 0, right?

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And then we take the derivative of the whole thing.

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And we just pretend like this whole expression, x-squared

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plus 2x plus 3 is just kind of like the variable x.

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We know that the derivative of x to the eighth is

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8x to the seventh.

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So it'll be 8 times something to the seventh.

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And that something is this entire expression here, 8 times

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x-squared plus 2x plus 3.

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I hope I didn't confuse you too much.

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And you can simplify this more in any way.

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Because it's 2x plus 2 times 8 times x-squared plus 2x

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plus 3 to the seventh.

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To multiply this out, or to multiply this out is

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a huge pain as you know.

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But we could simplify a little bit.

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Let me draw a divider here.

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We could say that that equals 8 times 2x, 16x plus 16 times--

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I'm making it really messy-- x-squared plus 2x plus 3

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to the seventh power.

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I hope I didn't confuse you too much.

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In the next presentation, I'm just going to do a ton of

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examples using the chain rule.

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And I think the more examples you see, it's going to

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hit the point home.

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And then after I've done a bunch of examples, then I'm

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going to give you a formal definition.

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I think that's actually an easier way to digest the chain

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rule than giving you the formal definition first, and then

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showing you a bunch of examples.

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So I'll see you in the next presentation.

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