is introduce ourselves

to the quotient rule.

And we're not going to prove it in this video.

In a future video we can prove it using the product rule

and we'll see it has some similarities to the product rule.

But here, we'll learn about what it is

and how and where to actually apply it.

So for example

if I have some function F of X

and it can be expressed as the quotient

of two expressions.

So let's say

U of X

over V of X.

Then the quotient rule tells us

that F prime of X

is going to be equal to

and this is going to look a little bit complicated

but once we apply it, you'll hopefully

get a little bit more comfortable with it.

Its going to be equal to the derivative

of the numerator function.

U prime of X.

Times the denominator function.

V of X.

Minus

the numerator function.

U of X.

Do that in that blue color.

U of X.

Times the derivative of the denominator function

times V prime of X.

And this already looks very similar to the product rule.

If this was U of X times V of X

then this is what we would get if we took the derivative

this was a plus sign.

But this is here, a minus sign.

But were not done yet.

We would then divide by

the denominator function squared.

V of X

squared.

So let's actually apply this idea.

So let's say that we have F of X

is equal to

X squared

over cosine of X.

Well what could be our U of X

and what could be our V of X?

Well, our U of X

could be our X squared.

So that is U of X

and U prime of X would be equal to two X.

And then this could be our V of X.

So this is V of X.

And V prime of X.

The derivative of cosine of X with respect to X

is equal to negative sine of X.

And then we just apply this.

So based on that

F prime of X

is going to be equal to

the derivative of the numerator function

that's two X, right over here, that's that there.

So it's gonna be two X

times the denominator function.

V of X is just cosine of X

times cosine of X.

Minus

the numerator function

which is just X squared.

X squared.

Times the derivative of the denominator function.

The derivative of cosine of X

is negative sine X.

So, negative

sine of X.

All of that over

all of that over

the denominator function squared.

So that's cosine of X

and I'm going to square it.

I could write it, of course, like this.

Actually, let me write it like that

just to make it a little bit clearer.

And at this point, we just have to simplify.

This is going to be equal to

let's see, we're gonna get

two X times cosine of X.

Two X

cosine of X.

Negative times a negative is a positive.

Plus, X squared

X squared

times sine of X.

Sine

of X.

All of that over

cosine of X squared.

Which I could write like this, as well.

And we're done.

You could try to simplify it, in fact,

there's not an obvious way to simplify this any further.

Now what you'll see in the future

you might already know something called

the chain rule, or you might learn it in the future.

But you could also do the quotient rule

using the product and the chain rule

that you might learn in the future.

But if you don't know the chain rule yet,

this is fairly useful.