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Now that you've seen some examples of the chain rule

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in use, I think the actual definition of the chain rule

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might be more digestible now.

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So let me give you the actual definition of the chain rule.

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Let's say I have a function f of x and it equals h of g of x.

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And you remember all this from composite functions.

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So the chain rule just says that the derivative of f of x

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or f prime of x is equal to the derivative of this inner

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function, g prime of x times the derivative of this h

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function, h prime of x.

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But it's not going to be just h prime of x.

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It's going to be h prime of g of x.

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So let's apply that to some examples like we were

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doing before, and I think it'll make some sense.

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So let's say we had f of x is equal to x squared plus 5x

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plus 3, all of this to the fifth power.

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So in this example, what's h of x, what's g of x, and

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you know what f of x is.

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Well let's say g of x would be this inner function.

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So we would say-- let me pick a different color-- g of x here

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is x squared plus 5x plus 3.

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It's the stuff, f g of x.

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Well h of g of x is this whole thing, so what would h of x be?

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This is h of g of x, but h of x would just be x

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to the fifth, right?

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Because this expression as you took this entire g of x and you

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put it in for x right here.

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I think that make sense if you take this entire expression and

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you substitute x here for this entire expression you

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get this expression.

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And this shows that this is equal to h of g of x.

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If you just take this blue part and substitute it for x, you

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get this entire expression.

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So the chain rule just tells us that the derivative of this,

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that f prime of x-- and I have a feeling I'm going to run out

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of space-- f prime of x-- well actually before I do anything,

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let's figure out the derivatives of g

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of x and h of x.

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g of x of g prime of x-- let me draw a little line here to

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divide it out, I know I'm running out of space.

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So g prime of x is equal to 2x plus 5.

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2x plus 5, and then derivative 3 is just 0, right?

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And the derivative of h of x? h prime of x is equal

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to 5 x to the fourth.

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So the chain rule just says that the derivative of this

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entire composite function is just-- let me just

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write it down here.

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I'm doing this to optimally confuse you.

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The derivative of this entire function is just g prime of x.

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Well we figured out with g prime of x is here, it's 2x

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plus 5 times h times h prime of g of x.

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So what's h prime of g of x?

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Well h prime of x is 5x to the fourth, but we want

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h prime of g of x.

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So h prime of g of x would equal 5 times g

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of x to the fourth.

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And we know what g of x is, it's this whole thing.

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So it would be times 5, and this whole thing x squared

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plus 5x plus 3, all that to the fourth power.

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I think I have truly, truly confused you, so I'm going

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to try to do a couple of more examples.

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Clear this.

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OK.

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Let me write it up here again.

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So if we say that f of x is equal to h of g of x, then f

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prime of x is equal to g prime of x times h prime of g of x.

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So I'll do another example.

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Let's say that g of x is equal to x to the seventh minus

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3x to the ninth is 3.

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And let's say that h of x is equal to-- let's do something

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reasonably straightforward.

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Let's say h of x is x to the minus 10.

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So what is f of x? f of x is just h of g of x, and this

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should be a bit of a reminder from composite functions.

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So let's see.

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h of g of x would just be-- you take g of x and you substitute

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it for x here, so it would just be this expression, x to the

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seventh minus 3x to the minus third, and then all of that

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to the minus 10th power.

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So this is our f of x.

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And this is of course equal to f of x, right, because f of

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x is equal to h of g of x.

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I know this very confusing, but bear with me.

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Maybe you have to watch the video twice and it'll

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start making more sense.

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Well we want to now figure out what f prime of x is.

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Well the chain rule tells us all it is, is we take the

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derivative of g of x, right?

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So the derivative of g of x is what?

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That's easy.

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Or hopefully it's easy by now.

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Derivative of g of x is 7x to the sixth, and minus 3 times

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minus 3 is plus 9x to the minus 4.

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I just took minus 3 and went down 1, so that's g prime of x.

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And then times h prime of g of x.

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Well what's h prime of x?

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That's easy.

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That's just minus 10 times x to the minus 11.

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But we want to do h prime of g of x.

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So instead of having an x here, we're going to substitute that

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x with the entire g of x expression.

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So this is just times 10 time something to the minus eleven,

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and that something is just g of x.

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x to the seventh, minus 3x access to the minus 3.

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And there's our answer. f prime of x is the derivative of kind

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of the inner function, g of x, times the derivative of the

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outer function, but instead of it just being applied to x it'd

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be applied to the entire g of x instead of an x being here.

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Maybe I've confused you more.

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Let me do one quick example just to show you that you

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don't have to kind of do this whole h of g of every time.

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So if I have f of x is equal to 5 times minus x to the eighth,

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plus x to the minus eighth, all of that over to

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the fifth power.

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If I want to figure out f prime of x I just take the derivative

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of this inner function I guess I could call it, so that's

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minus 8x to the seventh minus 8x-- because it's just take the

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negative 8-- to the minus ninth, times the derivative

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of this larger function.

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So 5 times 5 is 25 times something to the fourth.

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And that something is just going to be this expression

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minus x to the eighth plus x to the minus eighth.

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And we're done.

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You could simplify it.

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You could multiply this 25 out and do et cetera, et cetera.

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Hopefully this gives you more of an intuition of what the

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chain rule is all about, and I'm going to do a lot more

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examples in the next couple of presentations as well.

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See you soon.

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