Consider the functions f and

g with the graphs shown below. If capital G of x is equal to g

of f of x whole thing squared, what is the value of G

prime, capital G prime, of 5? And I encourage you to

now pause this video and try to solve it on your own. So let’s try to think

through this somewhat complicated-looking function

definition right over here. So we have capital G of x. And actually, let me

write it this way. Let me write it this way,

I’ll do it in yellow. We have capital G of x is

equal to this quantity squared. What we’re squaring

is g of f of x. g of f of x is what

we’re squaring. Or another way to

write G of x, If h of x were to be equal

to x squared, we could write G of x is equal

to h of this business, h of g of f of x. Let me just copy

and paste that so I don’t have to keep

switching colors. So copy and paste, there we go. So this is another

way of writing G of x, where

whatever g of f of x, we input then to h of x, which

is really just squaring it. So there’s a couple of

ways that we can write out the derivative of capital

G with respect to x. And you could

imagine this is going to involve the chain rule. But I like to write it

out, just to clarify in my head what’s going

on and to make sure that it actually

makes some sense. So one thing that

we could write, we could write the

derivative of G with respect– I’ll mix

notations a little bit– but I’ll write the derivative

of G of x with respect to x is equal to the

derivative of this whole thing. So let me copy and paste

it, copy and paste. It’s equal to this derivative

of this whole thing with respect to what’s inside of

that whole thing. So if you wanted to treat

g of f of x as a variable, so with respect to that. So copy and paste. So it’s going to be the

derivative of this whole thing with respect to g of f of

x times the derivative of g of f of x with respect to f

of x, with respect to– I’ll just copy and paste this

part, whoops– with respect to f of x. And I like to write this out. It feels good. It looks like these are rational

expressions with differentials. It’s really a notation more

than to be taken literally. But it feels good, or

at least in my mind it’s a little bit more intuitive

why all of this works out. So with respect to f of x times

the derivative of– and I’m using non-standard

notation here, but it helps me really

conceptualize this– times the derivative of f of

x with respect to x. Or another way we

could write this is G prime of x is

equal to h prime of g of f of x, h

prime of– actually, let me do it here–

h prime of this. So copy and paste,

h prime of that, times g prime of f of x,

times g prime of this. So copy and then paste. So times g prime of that. Put some parentheses there. Times f prime of x. And I like writing it this

way, because you notice if these were– and once

again, this is more notation, but it gives a sense

of what’s going on. If you did view

these as fractions, that would cancel with that. That would cancel with that. You’re taking the derivative

of everything with respect to x, which is exactly

what you wanted to do. And let me put some

parentheses here so it makes a little bit

clearer what’s going on. But this thing, in

my brain, I like to translate that

as, well, that’s just h prime of g of f of x. This is g prime of f of x. This is f prime of x. And going from this

to try to answer your question, the question

that they’re asking us actually isn’t too bad. So we want to know,

what’s G prime of 5? So everywhere we see an

x, let’s change it to a 5. So we’re going to say, we need

to figure out what G prime of 5 is. G prime of 5 is equal

to– and actually, let me just copy and

paste this whole thing. So copy and paste. And so let me, everywhere

where I see an x, I’m going to

replace it with a 5. So let me get rid of that. Let me get rid of that. And let me get rid of that. And so I have a 5, a 5, and a 5. So what is f of 5? f of 5 is equal to negative 1. So this right over here

simplifies to negative 1. This right over here

simplifies to negative 1. And what’s f prime

of negative 5? Well, that’s the slope

of the tangent line at this point right over here. And we see that the

derivative, or the slope, of the tangent line here is 0. So this right over here

is going to be equal to 0. Now that’s really interesting. So we could keep trying to,

well, what’s g of negative 1? What’s g prime of negative 1? You could see g of

negative 1, g of negative 1 we see is negative 1. g prime

of negative 1 is the slope here, which is also negative 1. Then we could calculate

h prime of these values, et cetera, et cetera. But we don’t even

have to do that. Because this is the

product of three things, and one of these things

right over here is a 0. So 0 times anything

times anything is going to be equal to 0. Another way of thinking

about it is, f of x is isn’t changing when

x is equal to 5. If f of x isn’t changing

when x is equal to 5, then the input into the g

isn’t going to be changing. So the g function isn’t going

to be– in the composition g of f of x– isn’t

going to be changing. And so h of g of f of x

isn’t going to be changing. So g of x isn’t

going to be changing. And so the derivative of

capital G of x at x equals 5 is going to be equal to 0.

So colorful and informative.

Taking A from B equals none because A starts the alphabet without A you don't have B

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Bone marrow cannon catches a planet as s war head and saves life after swimming in a volcano with your mind

Then you learn natural math as you enter s natural computer made from volcanic matters

Im a spider /have other cannons to fight

Common since good cannon bad cannon

All numbers end at 9

All letters end at O

Alpha beginning

Omega END

ABCDEFGHIJKLMNO

PQRSTUVWXYZ

ITS A CODE

APBQCRDSETFUGVHWIXJYKZ

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Set wix

Wow im the best now to figure out when and how to stop it

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I want some of that.

ewwwwwwwwwwwwwwwwwwwww

He's a Mathematician, not a Grammarian.

Changing colors all the time is very disturbing and cuts the flow…