WEBVTT



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Okay, our goal today is to find the derivative of functions like this. We know the

00:00:14.920 --> 00:00:18.640
derivative of sine, which is cosine.

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And we know the derivative of this polynomial, which is 9x squared minus 5.

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9x squared minus 5.

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But this is not a product of two functions, not a quotient of two functions, not a

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sum of two functions.

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What is this?

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This is called a composite function.

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We have a function inside another function that's called a composite function.

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We have some function inside another function.

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So in math, a composite function, you compose two functions. You put a function

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inside a

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function inside a function to make a new function.

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Okay.

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Alright, we often write like this

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f composed with g, which means

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f of g of

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x or something.

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Okay. Functions inside another function.

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So let me write the sine of 3x cubed minus 5x plus 6.

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So if my inside function is g of x, what is our g of x?

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3x cubed minus 5x plus 6.

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and then my outside function would be sine of X and then Y would be f of g of

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x. You plug in this everywhere you see an x and f, you get that composite function.

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So sometimes we refer to it as just the inside function and the outside function. So

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we need

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a rule called the composite function rule.

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let's I'll just tell you what it is. So the derivative of a composite function.

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It turns out it's the derivative of the outside function composed with the

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inside function and then you simply multiply by the derivative of the inside

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function.

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Let's prove this real quick. It's pretty easy.

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The derivative of this function.

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I could say y prime.

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How are we going to prove this?

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The limit as.

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Okay, we're actually not going to do this. We're going to use the other definition.

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What

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was the other definition of the derivative? F prime of a is the limit. X approaches

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a.

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f of x minus f of a, all over x minus a. That was the second definition I gave. It

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is easier with this definition.

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We use the limit as x approaches a of f of g of x minus f of g of a, divided by x

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minus a.

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f of g of x, our function, minus our function at the value a.

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And now we need a trick.

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Any guesses what the trick's going to be?

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Here's the trick. We're going to multiply. We're not going to add the number zero.

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We

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did that in the product in the quotient. Yeah. We're going to multiply by the

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number. What's

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It's the only thing you're allowed to multiply by. It doesn't change it. Number one.

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Okay, but a very special version of the number one.

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Now what?

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I'm just going to switch these two.

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And we're almost done.

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We're going to change the limit of a product to the product of two limits.

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This one should be obvious. What is this?

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That's the definition of the derivative of g.

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That is g prime of a.

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And then this is

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the definition

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of the derivative

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of f, but not at

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a.

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At g of a. This is like our

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value. Okay this is f prime of g of a.

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So as x goes to a, this goes to g of a.

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Or sorry, that goes to zero.

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Anyways, and there it is.

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That's the proof of the composite

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I'm going to remove q of x times b times x.

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So real quick, we know G prime of X is what, 9X squared minus 5?

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f prime of x is cosine of x.

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So this is f prime, but we plug in g of x in there.

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So f prime is this, but instead of x, it's g of x, which is this guy.

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Cosine of 3x cubed minus 5x plus 6, times g prime of x, which is 9x squared minus 5.

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And that is it.

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The derivative of the outside function with the inside plug in there times the

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derivative of the inside function. It is the composite function rule, also known as

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the chain rule.

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Why do we call it the chain rule? Why don't we call the composite function rule?

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You could chain it or does that mean chain it?

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You could chain it and make another function.

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Okay so the chain rule is the composite function rule.

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Again, the derivative of a composite function is the outside,

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derivative of the outside, composed with the inside, times the derivative of the

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inside.

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But it's often written like this. dy dx equals dy du times du dx.

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Okay, so going back to that previous example.

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Here, Y is a function of X, correct?

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I don't see any u's. Y is a function of x. Okay, so what in the world does dy, du

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mean? Okay,

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Our inside function is u: u equals 3x cubed minus 5x plus 6. Now y equals sine of u.

00:14:25.280 --> 00:14:46.709
So dy du, now y is a function of u. dy du is simply cosine of u. du dx, the

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derivative

00:14:47.780 --> 00:15:07.447
du over dx is 9x squared minus 5. So dy over du times du over dx is cosine u times

00:15:07.447 --> 00:15:11.380
9x squared minus 5.

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Except we do not want u in the final answer.

00:15:16.860 --> 00:15:18.420
So we put what u is.

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u is 3x cubed minus 5x plus 6.

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Okay.

00:15:54.220 --> 00:15:58.140
So, again, what does this have to do with chain rule or the word chain?

00:16:01.740 --> 00:16:14.235
If you have a function inside a function inside another function, okay, dy dx is

00:16:14.235 --> 00:16:18.400
going to be dy du

00:16:19.940 --> 00:16:33.600
times du dv times dv dx. Okay, it's a chain of a bunch of derivatives that are

00:16:33.600 --> 00:16:40.940
all connected by the inside variable. Okay, in fact, this notation, it looks

00:16:40.940 --> 00:16:47.400
like these all cancel and you end up with dy over dx. These aren't really

00:16:47.400 --> 00:16:53.540
fractions so it's not how it works. This is our derivative of our function y with

00:16:53.540 --> 00:16:59.440
respect to our variable u, derivative of our variable u with respect to variable v,

00:17:00.520 --> 00:17:05.532
derivative of our function v with respect to x. Okay and if there's a fourth

00:17:05.532 --> 00:17:06.200
function inside

00:17:06.200 --> 00:17:12.580
that you just multiply by the derivative of that and so on. It's a whole chain of

00:17:12.580 --> 00:17:21.700
derivatives that are all connected by the previous function or variable. So

00:17:21.700 --> 00:17:22.800
let's do some more examples.

00:17:23.800 --> 00:17:25.700
There's a composite function

00:17:26.580 --> 00:17:27.940
What's the inside function?

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Cosine of x. Sometimes I just put a u, call that u.

00:17:34.640 --> 00:17:37.960
You want to write it, u is cosine of x.

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And if u is cosine of x,

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what is our outside function?

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E to the u.

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Or e to the x, same thing.

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Okay, so, typically we don't mess with all these U's and stuff, but let me show you

00:18:08.855 --> 00:18:10.460
both ways here.

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We typically just do this, the derivative of our, what is derivative of E to the U?

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The same function, E to the U, you can either put that in there or you just type it

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in,

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or not typing any, write it in, times the derivative of the inside function,

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derivative of u is?

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Negative sine of x.

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We'd probably write it like this.

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Okay.

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dy dx.

00:18:52.240 --> 00:18:59.220
dx. Here dy du is e to the u.

00:19:01.240 --> 00:19:03.240
du dx is

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negative sine of x.

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This is this, e to the u,

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times du dx

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and then you substitute what u is.

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Let's do another one.

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What's my inside function?

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x to the fourth plus sine of x. I can call that u. So my outside

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function is either X to the fifth or U to the fifth. So what is the derivative of

00:20:42.100 --> 00:20:55.055
u to the fifth, 5u to the fourth. So x to the fourth plus sine of x. So derivative

00:20:55.055 --> 00:20:57.100
of the outside

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function, composed with the inside, times the derivative of the inside. We

00:21:02.466 --> 00:21:03.360
definitely need

00:21:03.360 --> 00:21:06.220
parentheses around that inside derivative; do not forget them.

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Is this a composite function?

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No.

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What rule do we use to find the derivative?

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The quotient rule.

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Okay, however, is this the same function? It is. Okay, now we would use the product

00:21:42.300 --> 00:21:48.400
rule because we have a product. And before we weren't able to do it this way because

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here we got a function inside a function. Okay, so because of the chain rule you

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don't ever have to use the quotient rule. You could just change it to a product.

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x to the fourth.

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x to the fourth.

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But now, when we find the derivative of this, we need the chain rule.

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What's my inside function?

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Sin of x. That's my u. Okay.

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So it's going to be derivative, or so my outside function is u to the negative 1

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power.

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What is the derivative of u to the negative 1?

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What rule we use for that?

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Uh oh.

00:23:23.400 --> 00:23:24.360
Power rule, thank you.

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This is going to be negative 1 u to the negative 2.

00:23:31.320 --> 00:23:38.596
So it's going to be negative 1 sine of x to the negative 2 times the derivative of

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u,

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derivative of sine of x which is cosine of x. If I clean this up a little bit,

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is that negative x to the fourth,

00:24:00.540 --> 00:24:05.340
cosine of x over sine of x squared.

00:24:10.760 --> 00:24:13.880
And if you made one fraction out of this,

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we have to multiply this by sine of x over sine of x.

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We get 4x cubed times sine of x minus x to the fourth times cosine of x, over sine

00:24:32.261 --> 00:24:33.280
squared x.

00:24:49.360 --> 00:24:54.532
Which is exactly, if you did the quotient rule, what you have got, the derivative of

00:24:54.532 --> 00:24:58.734
our first function times the second, or the denominator, minus the first function

00:24:58.734 --> 00:25:02.937
times the derivative of our denominator, divided by the derivatives, or sorry, the

00:25:02.937 --> 00:25:03.260
denominator.

00:25:03.260 --> 00:25:04.100
denominator square.

00:25:04.100 --> 00:25:07.100
Okay, so.

00:25:07.240 --> 00:25:11.665
So here's a composite function, but it's not a composite of two functions, it's a

00:25:11.665 --> 00:25:11.960
composite

00:25:11.960 --> 00:25:15.020
of three functions.

00:25:18.560 --> 00:25:19.460
Okay.

00:25:25.660 --> 00:25:34.240
If I call my very inside function u, let's call this u, no let's not do that.

00:25:35.960 --> 00:25:39.100
Let's call this whole inside u.

00:25:42.120 --> 00:26:00.340
We got y equals sine of u, correct? And then inside of u, sorry, so sine of u is

00:26:00.340 --> 00:26:23.600
u equals e to the v, and v is x squared minus 5x.

00:26:31.680 --> 00:26:33.840
Typically we don't need to do this.

00:26:35.040 --> 00:26:36.640
Ideally we just do this.

00:26:37.860 --> 00:26:41.920
Y prime. Derivative of our outside function which is sine.

00:26:42.860 --> 00:26:46.280
Cosine, with the same inside expression.

00:26:49.920 --> 00:26:52.120
Times now the derivative of the inside

00:26:52.920 --> 00:26:54.700
which is our same function.

00:26:56.400 --> 00:27:01.848
The derivative of the exponential is the same exponential times the derivative of

00:27:01.848 --> 00:27:05.200
its inside function, which is 2x minus 5.

00:27:10.840 --> 00:27:24.671
Let me show you using u and v. We want dy over dx. It is dy over du, because y is a

00:27:24.671 --> 00:27:25.421
function

00:27:25.660 --> 00:27:39.120
of u times du u is a function of v and v is a function of x.

00:27:47.640 --> 00:27:58.900
Okay, here dy du is cosine of u du dv,

00:27:58.900 --> 00:28:08.875
derivative of our function u with respect to v is e to the v dv dx derivative of v

00:28:08.875 --> 00:28:09.400
with

00:28:09.400 --> 00:28:28.360
respect to x is 2x minus 5. So this is cosine of u times this du dv is e to the

00:28:32.680 --> 00:28:48.300
times dv dx. And then simply we plug in

00:28:48.300 --> 00:29:06.723
what u and v are. u is e to the v, v is x squared minus 5x, and I actually need

00:29:06.723 --> 00:29:07.600
that.

00:29:14.380 --> 00:29:16.340
And then we get the same result.

00:29:17.900 --> 00:29:25.438
You got a function inside a function inside a function, you see, multiplying by the

00:29:25.438 --> 00:29:25.940
derivative

00:29:25.940 --> 00:29:27.000
of each inside function.

00:29:31.640 --> 00:29:33.460
Get questions on this so far.

00:29:33.460 --> 00:29:38.340
When do we need to use the chain rule?

00:29:38.640 --> 00:29:39.820
Only composite functions.

00:29:40.540 --> 00:29:43.260
Yeah, only composite functions.

00:29:43.700 --> 00:29:44.600
exponential.

00:29:46.060 --> 00:29:47.080
We don't know the derivative of this.

00:29:47.080 --> 00:29:56.080
we do know the derivative of e to the x which is e to the x. We don't know the

00:29:56.080 --> 00:30:06.863
derivative of 7 to the x. So how are we going to do this? We're definitely not going

00:30:06.863 --> 00:30:08.660
to use the

00:30:08.660 --> 00:30:24.040
definition of the derivative. Anybody got an idea? What were you gonna say?

00:30:24.040 --> 00:30:25.100
Oh, maybe.

00:30:30.660 --> 00:30:31.980
Do you know

00:30:31.980 --> 00:30:32.880
the derivative of the logs?

00:30:37.800 --> 00:30:40.140
We don't know the derivative of the natural log.

00:30:49.840 --> 00:30:51.980
Do we have a

00:30:51.980 --> 00:30:52.520
change of

00:30:52.520 --> 00:30:54.020
base formula for a logarithm?

00:30:54.020 --> 00:30:57.420
Do we have a change of base for exponentials?

00:31:31.500 --> 00:31:33.240
Is that the same function?

00:31:45.020 --> 00:31:49.480
What does e to the natural log of u equal?

00:31:50.640 --> 00:31:52.240
Simply U. Why is that?

00:31:57.040 --> 00:32:00.660
Because E to the X and LN of X are what?

00:32:00.660 --> 00:32:01.600
type of functions.

00:32:05.540 --> 00:32:06.800
So there's an I.

00:32:08.200 --> 00:32:12.880
Inverse functions. When you compose a function and it's inverse, you end up with

00:32:12.880 --> 00:32:15.060
back to what you started with.

00:32:17.660 --> 00:32:20.860
So e to the ln of seven to the x

00:32:20.860 --> 00:32:24.380
is the same thing as seven to the x. Regroup that?

00:32:26.440 --> 00:32:27.340
Okay.

00:32:30.180 --> 00:32:38.340
Does this make sense? Okay, and then rules of logarithms. What can I do with an

00:32:38.340 --> 00:32:39.360
exponent that's

00:32:39.380 --> 00:32:43.120
in a logarithm? We can take it out and

00:32:43.120 --> 00:33:05.560
it becomes the coefficient. Do we agree? Natural log of 7 is just a number. It is a

00:33:05.860 --> 00:33:11.360
constant. So let me write it one more time.

00:33:14.400 --> 00:33:17.100
e raised to the power x times natural log of 7.

00:33:20.120 --> 00:33:22.740
Okay. Now we're going to take the derivative of this.

00:33:24.380 --> 00:33:25.880
What is my inside function?

00:33:30.260 --> 00:33:33.520
Ln of seven. Sorry. Times x.

00:33:36.480 --> 00:33:45.680
So the derivative is going to be the derivative of e to the u, which is e to

00:33:45.680 --> 00:34:01.060
the u times the derivative, what is the derivative of the ln of 7 times x? Oh boy.

00:34:01.700 --> 00:34:09.260
what's the derivative of 5x? 5. What's the derivative of negative 8x? What's the

00:34:09.260 --> 00:34:16.037
The derivative of x times natural log of 7 is natural log of 7. This is a linear

00:34:16.037 --> 00:34:17.920
polynomial with a constant coefficient.

00:34:17.920 --> 00:34:29.660
number. Okay, it's weird because you guys, your logarithms are weird, but this is

00:34:29.660 --> 00:34:37.760
a polynomial with an unusual coefficient. The derivative of x times natural log of 7

00:34:37.760 --> 00:34:38.300
is

00:34:38.300 --> 00:34:49.664
ln of 7. Correct? Or am I just making up stuff? Okay, so this is the derivative of 7

00:34:49.664 --> 00:34:50.860
of the

00:34:50.860 --> 00:35:02.940
x. Let me remind you where we are. This is our function,

00:35:02.940 --> 00:35:18.220
7 to the x. Its derivative is the same function,

00:35:18.220 --> 00:35:23.900
times the natural logarithm of its base.

00:35:23.900 --> 00:35:32.300
there okay did it matter what the base was here what if I started with base 8

00:35:34.840 --> 00:35:47.400
8 8 8 8 8 the only thing it would change would be those two numbers okay

00:35:49.720 --> 00:35:59.286
So the derivative of any exponential function, let's call b to the x, and the base

00:35:59.286 --> 00:36:06.460
has to be greater than 0 and the base cannot equal 1.

00:36:09.220 --> 00:36:14.944
The derivative is simply the same exact function, just like e to the x, except we

00:36:14.944 --> 00:36:15.660
multiply it

00:36:15.660 --> 00:36:17.220
by the natural log of the base.

00:36:20.140 --> 00:36:25.160
Okay, it was because the chain rule we're able to find the derivative of this.

00:36:26.880 --> 00:36:30.340
So this is the derivative of any exponential function.

00:36:31.520 --> 00:36:33.880
What's the derivative of this?

00:36:39.780 --> 00:36:46.510
First off, constant multiple rule. It's just 5 times the derivative of 10 to the x.

00:36:46.510 --> 00:36:53.240
The derivative of 10 to the x is 10 to the x times ln of 10.

00:36:53.740 --> 00:36:57.580
Okay, we're not using any chain rule here. We just use the chain rule to prove the

00:36:57.580 --> 00:37:01.383
derivative of that exponential function. We now know the derivative of any

00:37:01.383 --> 00:37:01.700
exponential

00:37:01.700 --> 00:37:10.500
function, not just e to the x. Let us do another example.

00:37:12.440 --> 00:37:14.460
So this is a composite function.

00:37:15.460 --> 00:37:16.960
What's the inside function?

00:37:18.540 --> 00:37:18.840
Secant.

00:37:18.840 --> 00:37:34.600
and the outside function is 8 to the x or 8 to the u so the derivative of 8 to

00:37:34.600 --> 00:37:48.515
the u is 8 to the u times ln of 8. Okay, but u is secant of x times the derivative

00:37:48.515 --> 00:37:49.840
of the

00:37:50.070 --> 00:38:11.291
The derivative of secant x is secant x times tangent x. If the input were something

00:38:11.291 --> 00:38:15.270
other than x,

00:38:15.430 --> 00:38:24.049
it was like x cubed. The only thing with change would be secant of x cubed, tangent

00:38:24.049 --> 00:38:25.570
of x cubed,

00:38:25.570 --> 00:38:34.450
we would multiply by the derivative of x cubed, which is 3x squared.

00:38:34.450 --> 00:38:43.585
The derivative of the outside function times the derivative of the inside function,

00:38:43.585 --> 00:38:44.990
continuing inward.

00:38:44.990 --> 00:38:50.130
This is one of our most important and valuable rules. Now we can differentiate

00:38:50.130 --> 00:38:58.989
any kind of wacky function. Power rule, product rule, and any composition of

00:38:58.989 --> 00:38:59.739
functions.