F of negative one is three.

And F prime of negative one is equal to five.

Let G be the function G of X

is equal to one over X.

Let capital F be a function defined as

the product of those other two functions.

What is capital F prime of negative one?

Well, we can just apply the product rule here,

let me just rewrite let me just

essentially state the product rule.

Capital F prime of X

is going to be equal to

since capital F of X is the product

of these two functions,

when we apply the product rule this is gonna be

F prime of X

times G of X

plus plus

F of X

times G prime of X.

And so if we want to evaluate this

at F of negative one

capital F prime

at negative one is equal to

F prime of negative one

times G of negative one

plus function F evaluated at negative one

times the derivative of G

evaluated at negative one.

And let's see if we can figure these things out.

So do they tell us this anywhere?

Can we figure this out?

F prime of negative one.

Well they tell us right over here

F prime of negative one

is equal to five.

So this is equal to five.

Now, let's actually stick with F.

What is F of negative one?

Well, they tell that to us

right over here.

F of negative one is equal to three.

So F of negative one is equal to three.

Now G of negative one

and G prime of negative one

they don't give it to us explicitly here

but we can figure it out.

We can, we know that if G of X

is equal to this

G of negative one is equal

to one over negative one

which is equal to negative one.

So this is equal to

negative one.

And then last but not least

if we want to find G prime of negative one

we just have to take the derivative of this.

So G prime of X.

Actually just let me rewrite G of X.

G of X

one over X is just the same thing

as X to the negative one.

So we're gonna use a power rule

to figure out G prime of X

is equal to bring that exponent

out front, negative one times X

to the and then decrement the exponent

negative two power.

So, G prime of

negative one

of negative one is equal to

negative one times

negative one to the

negative two power.

And that's just the same thing as

negative one over negative one squared.

This is one.

So this is just all going to evaluate

to negative one.

So this is

negative one.

And so we have

five times negative one

which is negative five.

Plus three times negative one

which is negative three.

Which is equal to

negative eight.

So F prime of negative one

is equal to negative eight.