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

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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.

95
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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

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du over dx is 9x squared minus 5. So dy over du times du over dx is cosine u times

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9x squared minus 5.

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Except we do not want u in the final answer.

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So we put what u is.

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u is 3x cubed minus 5x plus 6.

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Okay.

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So, again, what does this have to do with chain rule or the word chain?

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If you have a function inside a function inside another function, okay, dy dx is

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going to be dy du

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times du dv times dv dx. Okay, it's a chain of a bunch of derivatives that are

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all connected by the inside variable. Okay, in fact, this notation, it looks

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like these all cancel and you end up with dy over dx. These aren't really

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fractions so it's not how it works. This is our derivative of our function y with

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respect to our variable u, derivative of our variable u with respect to variable v,

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derivative of our function v with respect to x. Okay and if there's a fourth

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function inside

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that you just multiply by the derivative of that and so on. It's a whole chain of

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derivatives that are all connected by the previous function or variable. So

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let's do some more examples.

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There's a composite function

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What's the inside function?

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

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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

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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.

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dx. Here dy du is e to the u.

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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

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u to the fifth, 5u to the fourth. So x to the fourth plus sine of x. So derivative

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of the outside

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

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definitely need

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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

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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.

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Power rule, thank you.

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

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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,

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cosine of x over sine of x squared.

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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

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squared x.

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Which is exactly, if you did the quotient rule, what you have got, the derivative of

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our first function times the second, or the denominator, minus the first function

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times the derivative of our denominator, divided by the derivatives, or sorry, the

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denominator.

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denominator square.

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Okay, so.

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So here's a composite function, but it's not a composite of two functions, it's a

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composite

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of three functions.

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Okay.

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If I call my very inside function u, let's call this u, no let's not do that.

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Let's call this whole inside u.

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We got y equals sine of u, correct? And then inside of u, sorry, so sine of u is

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u equals e to the v, and v is x squared minus 5x.

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Typically we don't need to do this.

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Ideally we just do this.

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Y prime. Derivative of our outside function which is sine.

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00:26:42,860 --> 00:26:46,280
Cosine, with the same inside expression.

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Times now the derivative of the inside

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which is our same function.

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The derivative of the exponential is the same exponential times the derivative of

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00:27:01,848 --> 00:27:05,200
its inside function, which is 2x minus 5.

201
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

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00:27:24,671 --> 00:27:25,421
function

203
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.

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00:27:47,640 --> 00:27:58,900
Okay, here dy du is cosine of u du dv,

205
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

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00:28:08,875 --> 00:28:09,400
with

207
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

208
00:28:32,680 --> 00:28:48,300
times dv dx. And then simply we plug in

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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

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00:29:06,723 --> 00:29:07,600
that.

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00:29:14,380 --> 00:29:16,340
And then we get the same result.

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00:29:17,900 --> 00:29:25,438
You got a function inside a function inside a function, you see, multiplying by the

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00:29:25,438 --> 00:29:25,940
derivative

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00:29:25,940 --> 00:29:27,000
of each inside function.

215
00:29:31,640 --> 00:29:33,460
Get questions on this so far.

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00:29:33,460 --> 00:29:38,340
When do we need to use the chain rule?

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00:29:38,640 --> 00:29:39,820
Only composite functions.

218
00:29:40,540 --> 00:29:43,260
Yeah, only composite functions.

219
00:29:43,700 --> 00:29:44,600
exponential.

220
00:29:46,060 --> 00:29:47,080
We don't know the derivative of this.

221
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

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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

223
00:30:06,863 --> 00:30:08,660
to use the

224
00:30:08,660 --> 00:30:24,040
definition of the derivative. Anybody got an idea? What were you gonna say?

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00:30:24,040 --> 00:30:25,100
Oh, maybe.

226
00:30:30,660 --> 00:30:31,980
Do you know

227
00:30:31,980 --> 00:30:32,880
the derivative of the logs?

228
00:30:37,800 --> 00:30:40,140
We don't know the derivative of the natural log.

229
00:30:49,840 --> 00:30:51,980
Do we have a

230
00:30:51,980 --> 00:30:52,520
change of

231
00:30:52,520 --> 00:30:54,020
base formula for a logarithm?

232
00:30:54,020 --> 00:30:57,420
Do we have a change of base for exponentials?

233
00:31:31,500 --> 00:31:33,240
Is that the same function?

234
00:31:45,020 --> 00:31:49,480
What does e to the natural log of u equal?

235
00:31:50,640 --> 00:31:52,240
Simply U. Why is that?

236
00:31:57,040 --> 00:32:00,660
Because E to the X and LN of X are what?

237
00:32:00,660 --> 00:32:01,600
type of functions.

238
00:32:05,540 --> 00:32:06,800
So there's an I.

239
00:32:08,200 --> 00:32:12,880
Inverse functions. When you compose a function and it's inverse, you end up with

240
00:32:12,880 --> 00:32:15,060
back to what you started with.

241
00:32:17,660 --> 00:32:20,860
So e to the ln of seven to the x

242
00:32:20,860 --> 00:32:24,380
is the same thing as seven to the x. Regroup that?

243
00:32:26,440 --> 00:32:27,340
Okay.

244
00:32:30,180 --> 00:32:38,340
Does this make sense? Okay, and then rules of logarithms. What can I do with an

245
00:32:38,340 --> 00:32:39,360
exponent that's

246
00:32:39,380 --> 00:32:43,120
in a logarithm? We can take it out and

247
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

248
00:33:05,860 --> 00:33:11,360
constant. So let me write it one more time.

249
00:33:14,400 --> 00:33:17,100
e raised to the power x times natural log of 7.

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Okay. Now we're going to take the derivative of this.

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What is my inside function?

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Ln of seven. Sorry. Times x.

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So the derivative is going to be the derivative of e to the u, which is e to

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the u times the derivative, what is the derivative of the ln of 7 times x? Oh boy.

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what's the derivative of 5x? 5. What's the derivative of negative 8x? What's the

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The derivative of x times natural log of 7 is natural log of 7. This is a linear

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polynomial with a constant coefficient.

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number. Okay, it's weird because you guys, your logarithms are weird, but this is

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a polynomial with an unusual coefficient. The derivative of x times natural log of 7

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is

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ln of 7. Correct? Or am I just making up stuff? Okay, so this is the derivative of 7

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of the

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x. Let me remind you where we are. This is our function,

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7 to the x. Its derivative is the same function,

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times the natural logarithm of its base.

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there okay did it matter what the base was here what if I started with base 8

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8 8 8 8 8 the only thing it would change would be those two numbers okay

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So the derivative of any exponential function, let's call b to the x, and the base

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has to be greater than 0 and the base cannot equal 1.

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The derivative is simply the same exact function, just like e to the x, except we

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multiply it

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by the natural log of the base.

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Okay, it was because the chain rule we're able to find the derivative of this.

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So this is the derivative of any exponential function.

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What's the derivative of this?

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First off, constant multiple rule. It's just 5 times the derivative of 10 to the x.

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The derivative of 10 to the x is 10 to the x times ln of 10.

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Okay, we're not using any chain rule here. We just use the chain rule to prove the

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derivative of that exponential function. We now know the derivative of any

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exponential

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function, not just e to the x. Let us do another example.

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So this is a composite function.

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What's the inside function?

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Secant.

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and the outside function is 8 to the x or 8 to the u so the derivative of 8 to

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the u is 8 to the u times ln of 8. Okay, but u is secant of x times the derivative

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of the

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The derivative of secant x is secant x times tangent x. If the input were something

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other than x,

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it was like x cubed. The only thing with change would be secant of x cubed, tangent

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of x cubed,

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we would multiply by the derivative of x cubed, which is 3x squared.

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The derivative of the outside function times the derivative of the inside function,

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continuing inward.

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This is one of our most important and valuable rules. Now we can differentiate

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any kind of wacky function. Power rule, product rule, and any composition of

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functions.