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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,