Chain rule
The Chain Rule Part 4 of derivatives. Introduction to the chain rule.
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- Welcome back.
- I'm now going to do some more examples of a bit of a review
- of some of the derivatives that we've been seeing.
- And then I'll introduce you to something called the chain rule
- which expands the universe of the types of functions we can
- take the derivatives of.
- So in the last presentation, I showed you how to function if I
- had f of x is equal to 10x to the seventh plus 6x to the
- third plus 15x minus x to the 16th.
- To take the derivative of this entire function, we take just
- the derivatives of each of the pieces, right?
- Because you can add them up.
- So f prime of x in this example, is equal to-- and
- I think you get the hang of it at this point.
- It's actually fairly straightforward.
- We take the 7, multiply it by the 10.
- So we get 70x, then 1 degree less.
- So 70x to the sixth plus 18x squared plus 15.
- We can kind of view this as x to the 1, right?
- So it's 1 times 15 times x to the 0.
- Which is 1.
- So that's just 15 minus 16x to the 15th.
- And I don't want you to lose sight of what we're
- actually doing here.
- What is f prime of x?
- This is the function the tells us the slope of any point
- x, along the curve f of x.
- It's a pretty interesting thing.
- Let me just draw to maybe give you a little bit of intuition.
- 32 00:01:45,3 --> 00:01:47,49 I don't know what the slope of f of x really looks like.
- And actually, let's pretend like this isn't f of x.
- Let's pretend like this is just some arbitrary
- function I'm drawing.
- If this is f of x, just some curve that does all sorts of
- crazy things, f prime of x tells me the slope at any
- point along that line.
- So if I wanted to know the slope at this point right here,
- I could use the derivative function to figure out the
- slope of the tangent line.
- The tangent line is something like that right there.
- Or if I wanted to figure out the slope at this point,
- once again I'd use the derivative function.
- And it would tell me the slope of the tangent
- line at that point.
- Which would be something like that.
- So it's a pretty useful thing.
- And once I give you all the tools to analytically solve a
- whole host of derivatives, then we'll actually do a bunch
- of word problems and applications of derivatives.
- And I think you'll see that it's a really, really,
- really useful concept.
- So let's move on.
- I think you get the idea of how to do these derivatives
- of polynomials.
- Let me erase this.
- I'm actually using a different tool now.
- So I think it might be a bit easier.
- Let's see, someone was calling me.
- But you're more important so I will not answer the phone.
- 63 00:03:08,44 --> 00:03:11,64 I'm going to introduce you-- this tool doesn't have, I
- don't think it has a straight up eraser.
- Actually, maybe let's see.
- If I do it like this.
- 68 00:03:27,48 --> 00:03:28,4 Oh let me see.
- 70 00:03:30,9 --> 00:03:32,15 No that doesn't work.
- Let me just erase like this, the old-fashioned way.
- 73 00:03:39,53 --> 00:03:41 You just have to bear with me.
- This is good.
- It feels like I'm a real teacher with a real chalkboard
- and a real eraser now.
- 79 00:03:53,53 --> 00:03:56,75 This is a lot cleaner than a normal chalkboard as well.
- Bear with me, almost there.
- I'll figure out a faster way to do this over the
- next couple of videos.
- It's pretty sad.
- I'm showing you how to do derivatives in calculus, but
- I don't know how to erase a faster way than this.
- 87 00:04:14,27 --> 00:04:15,56 There, we're done.
- OK.
- So now I'm going show you how to solve the derivatives of a
- slightly more complicated type of a function.
- It's actually not more complicated.
- It's just different.
- So let's say f of x is equal to 2x plus 3 to the fifth power.
- And I want to figure out the derivative of this.
- We're going to use something called the chain rule.
- Because one thing we could do, we could just multiply out 2x
- plus 3 to the fifth power.
- And if you've ever done that, you know it's a pain.
- So that's not something we'd want to do.
- So we're going to use something called the chain rule.
- And I'm just going to give you a bunch of examples before I
- even show you the definition of the chain rule.
- Because I think this is something that you just
- have to learn by example.
- So the chain rule just tells us that the derivative of let's
- say this function right here.
- You take the derivative of the subfunctions, and then you can
- But I think when you introduce it formally, it gets
- more confusing.
- So what I do, I just take the derivative of 2x plus 3 first.
- So I take the derivative of 2x plus 3.
- What's the derivative of 2x plus 3?
- Well you know that.
- It's just the derivative of 2x, which is 2.
- And then the derivative of 3 is 0.
- So the derivative of 2x plus 3 is just 2.
- And then I'm going to multiply that times the derivative
- of the whole thing.
- And I just pretend like 2x plus 3 is just like
- a variable by itself.
- So then what's the derivative of x to the fifth?
- Well the derivative of x to the fifth-- I'm going to do that in
- a different color-- the derivative of x to the
- fifth is 5x to the fourth.
- So it'll be 5 times something to the fourth.
- But here we didn't take the derivative of x the fifth.
- We took the derivative of 2x plus 3 to the fifth.
- So we just put the 2x plus 3 there instead.
- 134 00:06:24,23 --> 00:06:26,19 So what did we do here?
- We went in the inside of the function, and we took
- the derivative here.
- And the derivative of 2x plus 3 was just 2.
- And then we multiplied it by the derivative of
- the greater function.
- And we just pretended like the 2x plus 3 was a variable.
- It was like x.
- So instead of 5x to the fourth, we got 5 times
- 2x plus 3 to the fourth.
- And if we just simplify that, f prime of x is equal to
- 2 times 5 is 10; 10 times 2x plus 3 to the fourth.
- That was a lot simpler than multiplying out 2x plus 3 to
- the fifth power, and then doing the derivatives the old way.
- I know this was probably a little confusing to you,
- so I'm going to try to do a couple more examples.
- Let's say I had g of x is equal to x-squared plus 2x plus
- 3 to the eighth power.
- So g prime of x is going to equal-- well what did we say?
- We take the derivative of the inside.
- This is called the chain rule.
- What's the derivative of the inside?
- It's 2x plus 2 plus 0, right?
- And then we take the derivative of the whole thing.
- And we just pretend like this whole expression, x-squared
- plus 2x plus 3 is just kind of like the variable x.
- We know that the derivative of x to the eighth is
- 8x to the seventh.
- So it'll be 8 times something to the seventh.
- And that something is this entire expression here, 8 times
- x-squared plus 2x plus 3.
- I hope I didn't confuse you too much.
- And you can simplify this more in any way.
- Because it's 2x plus 2 times 8 times x-squared plus 2x
- plus 3 to the seventh.
- To multiply this out, or to multiply this out is
- a huge pain as you know.
- But we could simplify a little bit.
- Let me draw a divider here.
- We could say that that equals 8 times 2x, 16x plus 16 times--
- I'm making it really messy-- x-squared plus 2x plus 3
- to the seventh power.
- I hope I didn't confuse you too much.
- In the next presentation, I'm just going to do a ton of
- examples using the chain rule.
- And I think the more examples you see, it's going to
- hit the point home.
- And then after I've done a bunch of examples, then I'm
- going to give you a formal definition.
- I think that's actually an easier way to digest the chain
- rule than giving you the formal definition first, and then
- showing you a bunch of examples.
- So I'll see you in the next presentation.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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