derivative of this G of X

and at first it can look intimidating.

We have a sine of X here.

We have a cosine of X.

We have this crazy expression here

with a pi over cube root of X

we're squaring the whole thing

and at first it might seem intimidating.

But as we'll see in this video, we can

actually do this with the tools already in our tool kit.

Using our existing derivative properties

using what we know about the power rule

which tells us the derivative with respect to X.

Of X to the N

is equal to N

times X to the N minus one,

we've see that multiple times.

We also need to use the fact

that the derivative of cosine of X

is equal to negative the sine of X.

And the other way around

the derivative with respect to X

of sine of X

is equal to positive

cosine of X.

So using just that we can actually evaluate this.

Or evaluate G prime of X.

So, pause the video and see if you can do it.

So probably the most intimidating part of this

because we know the derivative's a sine

of X and cosine of X is this expression here.

And we can just rewrite this

or simplify it a little bit so it takes a form

that you might be a little bit more familiar with.

So, so let me just do this

on the side here.

So, pi

pi over

the cube root of X

squared.

Well that's the same thing.

This is equal to pi squared

over the cube root of X squared.

This is just exponent properties

that we're dealing with.

And so this is the same thing.

We're gonna take X to the one third power

and then raise that to the second power.

So this is equal to pi squared

over.

Let me write it this way,

I'm not gonna skip any steps

because this is a good review of exponent property.

X to the one third squared.

Which is the same thing as pi squared

over X to the two thirds power.

Which is the same thing as pi squared

times X to the negative two thirds power.

So when you write it like this

it starts to get into a formula,

you're like, oh, I can see how the power rule

could apply there.

So this thing is just pi squared

times X to the negative two thirds power.

So actually let me delete this.

So,

this thing

can be rewritten.

This thing can be rewritten

as pi squared

times X to the negative

to the negative two thirds power.

So now let's take the derivative

of each of these pieces of this expression.

So, we're gonna take

we want to evaluate what the G prime of X is.

So G prime of X

is going to be equal to.

You could view it as a derivative with respect

to X of seven sine of X.

So

we can take do the derivative operator

on both sides here just to make it clear

what we're doing.

So we're gonna apply it there.

We're gonna apply it there.

And we're going to apply it

there.

So this derivative

this is the same thing as

this is going to be seven times the derivative

of sine of X.

So this is just gonna be seven times

cosine of X.

This one, over here,

this is gonna be three, or we're subtracting,

so it's gonna be this subtract

this minus.

We can bring the constant out

that we're multiplying the expression by.

And the derivative of cosine of X

so it's minus three times

the derivative of cosine of X

is negative sine of X.

Negative sine of X.

And then finally

here in the yellow we just apply the power rule.

So, we have the negative two thirds,

actually, let's not forget this minus sign

I'm gonna write it out here.

So you have the negative two thirds.

You multiply the exponent times the coefficient.

It might look confusing, pi squared,

but that's just a number.

So it's gonna be

negative

and then you have negative two thirds

times pi squared.

Times pi squared.

Times X to the negative two thirds

minus one power.

Negative two thirds

minus one power.

So what is this going to be?

So we get G prime of X

is equal to

is equal to

seven cosine of X.

And let's see, we have a negative three

times a negative sine of X.

So that's a positive three sine of X.

And then

we have, we're subtracting

but then this is going to be a negative,

so that's going to be a positive.

So we can say

plus two pi squared over three.

Two pi squared over three.

That's that part there.

Times X

to the.

So negative two thirds

minus one,

we could say negative one and two thirds,

or we could say negative five thirds by power.

Negative five thirds power.

And there you have it.

We were able to tackle this thing that

looked a little bit hairy

but all we had to use was the power rule

and what we knew to be the derivatives

of sine and cosine.