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  • - [Voiceover] Let's say we have the function f of x

  • which is equal to cosine of x to the third power

  • which we could also write like this,

  • cosine of x to the third power.

  • And we are interested in figuring out

  • what f prime of x is going to be equal to.

  • So we want to figure out f prime of x and as we will see,

  • the chain rule is going to be very useful here

  • and what I'm going to do is

  • I'm going to first just apply the chain rule

  • and then maybe dig into it a little bit

  • to make sure we draw the connection

  • between what we're doing here and then what you might see

  • in maybe some of your Calculus textbooks

  • that explain the chain rule.

  • So if we have a function

  • that is defined as essentially a composite function,

  • notice this expression right here,

  • we are taking something to the third power.

  • It isn't just an x that we're taking to the third power.

  • We are taking a cosine of x to the third power.

  • So we're taking a function, you could view it this way,

  • we're taking the function cosine of x

  • and then we're inputting it in to another function

  • that takes it to the third power.

  • So let me put it this way.

  • If you viewed,

  • if you say, look, we could take an x,

  • we put it into one function and that is,

  • that first function is cosine of x

  • so first, we evaluate the cosine

  • and so that's going to produce cosine of x,

  • cosine of x,

  • and then we're going to input it into a function

  • that just takes things to the third power.

  • So it just takes things to the third power.

  • And so what are you going to end up with?

  • Well, you're going to end up with,

  • what are you taking to the third power?

  • You're taking cosine of x.

  • Cosine of x to the third power.

  • This is a composite function.

  • You could view this,

  • you could view this as the function,

  • let's call this blue one, the function v

  • and let's call this the function u

  • and so if we're taking x and into u,

  • this is u of x

  • and then if we're taking u of x into the input

  • or as the input into the function v

  • then this output right over here,

  • this is going to be v of,

  • well, what was inputted?

  • V of u of x.

  • V of u of x

  • or another way of writing it,

  • I'm going to write it multiple ways.

  • That's the same thing as v of cosine of x.

  • V of cosine of x.

  • And so v, whatever you input into it,

  • it just takes it to the third power.

  • If you were to write v of x,

  • it would be x to the third power.

  • So the chain rule tells us

  • or the chain rule is what our brain should say.

  • Hey, it becomes applicable

  • if we're going to take the derivative of a function

  • that can be expressed as a composite function like this.

  • So just to be clear, we can write f of x.

  • f of x is equal to v of u of x.

  • I know I'm essentially saying the same thing

  • over and over again

  • but I'm saying it in slightly different ways

  • because the first time you learn this,

  • it can be a little bit hard to grok

  • or really deeply understand

  • so I'm going to try to write it in different ways.

  • And the chain rule tells us

  • that if you have a situation like this

  • then the derivative, f prime of x,

  • and this is something that you will see in your textbooks.

  • Well, this is going to be

  • the derivative of this whole thing

  • with respect to u of x

  • so we could write that as v prime of u of x.

  • V prime of u of x

  • times the derivative of u with respect to x.

  • Times u prime of x.

  • This right over here,

  • this is one expression of the chain rule

  • and so how do we evaluate it in this case?

  • Well, let me color code it in a similar way.

  • So the v function,

  • this outer thing that just takes things to the third power,

  • I'll put in blue.

  • So f prime of x,

  • another way of expressing it

  • and I'll use it with more of the differential notation,

  • you could view this as the derivative of,

  • well, I'll write it a couple of different ways.

  • You could view it as the derivative of v.

  • The derivative of v

  • with respect to u.

  • I want to get the colors right.

  • The derivative of v with respect to u,

  • that's what this thing is right over here,

  • times the derivative of u

  • with respect to x.

  • So times the derivative of u with respect to x.

  • And just to be clear,

  • so you're familiar with the different notations

  • you'll see in different textbooks,

  • this is this right over here just using different notations

  • and this is this right over here.

  • So let's actually evaluate these things.

  • You're probably tired of just talking in the abstract.

  • So this is going to be equal to,

  • this is going to be equal to

  • and I'm going to write it out again,

  • this is the derivative,

  • instead of just writing v and u,

  • I'm going to write it, let me write this way.

  • This is going to be,

  • I keep wanting, I'm using the wrong colors.

  • This is going to be the derivative of,

  • and I'm going to leave some space,

  • times the derivative of something else

  • with respect to something else

  • so we're going to have to first take the derivative of v.

  • Well, v is

  • cosine of x to the third power.

  • Cosine of x.

  • We're going to take the derivative of that

  • with respect to u which is just cosine of x

  • and we're going to multiply that

  • times the derivative of u which is cosine of x

  • with respect to x.

  • With respect to x.

  • So this one, we have good,

  • we've seen this before.

  • We know that the derivative with respect to x

  • of cosine of x.

  • Cosine.

  • We use it in that same color.

  • The derivative of cosine of x,

  • well, that's equal to negative sine of x.

  • So this one right over here, that is negative sine of x.

  • You might be more familiar with seeing

  • the derivative operated this way

  • but in theory, you won't see this as often

  • but this helps my brain really grok what we're doing.

  • We're taking the derivative of cosine of x

  • with respect to x.

  • Well, that's going to be negative sine of x.

  • Well, what about taking the derivative

  • of cosine of x to the third power

  • with respect to cosine of x?

  • What is this thing over here mean?

  • Well, if I were taking the derivative,

  • if I was taking the derivative of,

  • let me write it this way,

  • if I was taking the derivative of x to the third power,

  • x to the third power with respect to x,

  • if it was like that,

  • well, this is just going to be

  • and let me put some brackets here

  • to make it a little bit clear.

  • If I'm taking the derivative of that,

  • that is going to be,

  • that is going to be,

  • we bring the exponent out front.

  • That's going to be three,

  • three times x.

  • Three times x to the second power.

  • Three times x to the second power.

  • So the general notion here is

  • if I'm taking the derivative of something,

  • whatever this something happens to be,