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I'm now going to do a bunch more examples

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using the chain rule.

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So let's see.

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Once again.

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If I had f of x is equal to, let's see, I don't like this

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tool that I'm using now, let's have one of these.

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f of x is equal to, say, x to the third plus 2x squared

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minus, let's say, minus x to the negative 2.

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We haven't put any negative exponents in yet, but I

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think you'll see that the same patterns apply.

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And all of that to, let's say, the minus seven.

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We want to figure out what f prime of x, what the

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derivative of f of x is.

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So this might seem very complicated and daunting to

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you, and obviously to take this entire polynomial to

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the negative seventh power would take you forever.

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But using the chain rule, we can do it quite quickly.

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So the first thing we want to do, is we want to take the

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derivative of the inner function, I guess

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you could call it.

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We want to take the derivative of this.

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And what's the derivative of x to the third plus 2x squared

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minus x to the negative 2?

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Well, we know how to do that.

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That was the first type of derivatives we

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learned how to do.

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It's 3x squared and 2 times 2, plus 4x to the first, or just

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4x, and then here, with a negative exponent, we do

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the exact same thing.

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We say negative 2 times negative 1, right, there's a 1

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here, we don't write it down.

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So negative 2 times negative 1 is plus 2 x to the, and then we

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decrease the exponent by 1, so it's x to the

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negative 3, right?

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So we figured out what the derivative of the inside is,

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and then we just multiply that, that whole thing, times

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the derivative of kind of the entire expression.

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So then that'll be, we take the minus 7, let me

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do a different color.

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So this is the entire thing.

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So then we take minus 7, so it's times minus 7, this

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whole expression, I'm going to run out of space.

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x to the third plus 2x squared minus x to the minus 2.

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That's minus x to the minus 2.

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And all of that, we just decrease this exponent

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by 1, to the minus 8.

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So let me write it all down a little bit neater now.

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So we get f prime of x as the derivative of f of x is equal

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to 3x squared plus 4x plus 2x to the minus third power, I

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don't know why did that.

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That's minus 3.

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Times minus seven times x to the third plus 2x squared minus

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x to the minus two, all of that to the negative eight power.

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And we could simplify it little bit.

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Maybe we could just multiply this minus 7, times, we could

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distribute it across this expression.

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So we'd say, that equals minus 7, so this equals minus 21 x

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squared, minus 28 x minus 14x to the negative 3.

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All of that times x to the third plus 2 x squared minus x

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to the minus 2 to the minus 8.

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So there we did it.

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We took this, what I would say is a very complicated function,

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and using the chain rule and just the basic rules we had

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introduced a couple of presentations ago, we were able

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to find the derivative of it.

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And now, if we wanted to, for whatever application, we could

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find the slope of this function at any point x by just

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substituting that point into this equation, and we'll get

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the slope at that point.

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Let me do a slightly harder one, to show you that the chain

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rule, you can kind of go arbitrarily deep in

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the chain rule.

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OK.

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So let's say I had, let me see if I can write it

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a little bit thinner.

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If I had f of x, I don't know if you can see that, I'm going

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to do it a little fatter.

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f of x is equal to, I want to make it a little bit more

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complicated this time.

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3x to the minus 2 plus 5 x to the third minus 7x, all of that

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to the fifth, and then this whole expression to

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the third power.

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So I imagine you saying, Sal, you're starting to go nuts,

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this is going to take us forever.

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Well, I'll show you, using the chain rule, it will

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not take that long.

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So the way I think about it, so-- f prime of x,

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f prime of x equals.

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I start off kind with the innermost function.

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So let me see if I can use colors to make it

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a little bit simpler.

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Let's take the derivative of this innermost function first.

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Actually, let me give you the big picture.

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We want to find the derivative of the innermost function, and

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then a little bit bigger, and then a little bit

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more big than that.

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I know that's not precise mathematical terms, but

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you'll get the point when I show you this example.

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So first we'll do this inner function, this

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inner expression.

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And the derivative of that's pretty easy, right?

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It's 15x squared minus 7, right?

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that was pretty straightforward.

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And now we're going to want to multiply that times this

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entire derivative here.

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So let me circle that in a different-- so then

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we want to do this.

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We're going to multiply that times this entire derivative.

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Well, that's just times 5.

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And we just pretend like this is just an x here, right?

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Because the derivative of x to the fifth is 5x

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to the fourth, right?

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But instead of an x, we have this whole expression, 5x

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to the third minus 7x.

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So we'll write that.

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5x to the third minus 7x.

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Now the exponent here goes down by one.

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So it's 5 times 5x to the third minus 7x, all that

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to the fourth power.

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So we figured out the derivative of this so far, and

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then we want to figure out the derivative of this, so we'll

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add it, right, because we're trying to figure the derivative

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of this entire expression.

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So this is an easy one.

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Let me draw that in a different color.

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So we want the derivative of this.

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So that's negative 2 times 3, so that's negative

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6x to the minus three.

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So what have we done so far?

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We've so far figured out the derivative of this entire

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expression, right?

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The derivative of that entire expression using

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the chain rule is this.

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And now, we're almost done.

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We just have to multiply that.

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So I'm going to just, I've run out of space on that line, but

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let's just assume that the line continues.

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So that's times.

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And now we just take the derivative of kind of

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this whole big thing.

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And now it's going to be the derivative of, I'm going

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to use this brown color.

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So it's a whole big expression to the third power, right?

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So that becomes times 3 times the whole expression, right?

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That's 3 times, now I'm going to write the whole thing, 3x to

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the minus 2 plus 5 x the third minus 7x, that to fifth, and

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then you decrement this by 1, to the second power.

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That was an ultraconfusing example, and this is probably

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the hardest chain rule problem you'll see in a lot of

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the questions you'll have on your test.

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You see, it wasn't that difficult.

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We just kind of went to the smallest possible function, and

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actually the smallest possible function would have been one of

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these terms, but we just found the derivative of this, which

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was 15 x squared minus 7, and then we just used the principle

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that the derivative of kind of a function is just the

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derivative of each of its parts-- well, actually, the

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derivative of-- we figured out the derivative of this inner

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piece, which was 15x squared minus 7, and then we multiplied

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it times the derivative of this slightly larger piece, which is

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5 times this entire expression to the fourth, then we added it

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to the derivative of 3x to the minus 2.

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And then that whole thing, and actually I should put a big

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parentheses around here, that whole thing, we multiply it

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times the derivative of this larger expression.

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I think I might have confused you, so I apologize if I have,

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and in the next presentation I'm going to just do a bunch

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more chain rule problems, and at some point, it should

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start to make sense to you.

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I think it's just a matter of seeing example, after

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example, after example.

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I'll see you into the next presentation, and I apologize

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if I have confused you.