Now that you've seen some examples of the chain rule
in use, I think the actual definition of the chain rule
might be more digestible now.
So let me give you the actual definition of the chain rule.
Let's say I have a function f of x and it equals h of g of x.
And you remember all this from composite functions.
So the chain rule just says that the derivative of f of x
or f prime of x is equal to the derivative of this inner
function, g prime of x times the derivative of this h
function, h prime of x.
But it's not going to be just h prime of x.
It's going to be h prime of g of x.
So let's apply that to some examples like we were
doing before, and I think it'll make some sense.
So let's say we had f of x is equal to x squared plus 5x
plus 3, all of this to the fifth power.
So in this example, what's h of x, what's g of x, and
you know what f of x is.
Well let's say g of x would be this inner function.
So we would say-- let me pick a different color-- g of x here
is x squared plus 5x plus 3.
It's the stuff, f g of x.
Well h of g of x is this whole thing, so what would h of x be?
This is h of g of x, but h of x would just be x
to the fifth, right?
Because this expression as you took this entire g of x and you
put it in for x right here.
I think that make sense if you take this entire expression and
you substitute x here for this entire expression you
get this expression.
And this shows that this is equal to h of g of x.
If you just take this blue part and substitute it for x, you
get this entire expression.
So the chain rule just tells us that the derivative of this,
that f prime of x-- and I have a feeling I'm going to run out
of space-- f prime of x-- well actually before I do anything,
let's figure out the derivatives of g
of x and h of x.
g of x of g prime of x-- let me draw a little line here to
divide it out, I know I'm running out of space.
So g prime of x is equal to 2x plus 5.
2x plus 5, and then derivative 3 is just 0, right?
And the derivative of h of x? h prime of x is equal
to 5 x to the fourth.
So the chain rule just says that the derivative of this
entire composite function is just-- let me just
write it down here.
I'm doing this to optimally confuse you.
The derivative of this entire function is just g prime of x.
Well we figured out with g prime of x is here, it's 2x
plus 5 times h times h prime of g of x.
So what's h prime of g of x?
Well h prime of x is 5x to the fourth, but we want
h prime of g of x.
So h prime of g of x would equal 5 times g
of x to the fourth.
And we know what g of x is, it's this whole thing.
So it would be times 5, and this whole thing x squared
plus 5x plus 3, all that to the fourth power.
I think I have truly, truly confused you, so I'm going
to try to do a couple of more examples.
Clear this.
OK.
Let me write it up here again.
So if we say that f of x is equal to h of g of x, then f
prime of x is equal to g prime of x times h prime of g of x.
So I'll do another example.
Let's say that g of x is equal to x to the seventh minus
3x to the ninth is 3.
And let's say that h of x is equal to-- let's do something
reasonably straightforward.
Let's say h of x is x to the minus 10.
So what is f of x? f of x is just h of g of x, and this
should be a bit of a reminder from composite functions.
So let's see.
h of g of x would just be-- you take g of x and you substitute
it for x here, so it would just be this expression, x to the
seventh minus 3x to the minus third, and then all of that
to the minus 10th power.
So this is our f of x.
And this is of course equal to f of x, right, because f of
x is equal to h of g of x.
I know this very confusing, but bear with me.
Maybe you have to watch the video twice and it'll
start making more sense.
Well we want to now figure out what f prime of x is.
Well the chain rule tells us all it is, is we take the
derivative of g of x, right?
So the derivative of g of x is what?
That's easy.
Or hopefully it's easy by now.
Derivative of g of x is 7x to the sixth, and minus 3 times
minus 3 is plus 9x to the minus 4.
I just took minus 3 and went down 1, so that's g prime of x.
And then times h prime of g of x.
Well what's h prime of x?
That's easy.
That's just minus 10 times x to the minus 11.
But we want to do h prime of g of x.
So instead of having an x here, we're going to substitute that
x with the entire g of x expression.
So this is just times 10 time something to the minus eleven,
and that something is just g of x.
x to the seventh, minus 3x access to the minus 3.
And there's our answer. f prime of x is the derivative of kind
of the inner function, g of x, times the derivative of the
outer function, but instead of it just being applied to x it'd
be applied to the entire g of x instead of an x being here.
Maybe I've confused you more.
Let me do one quick example just to show you that you
don't have to kind of do this whole h of g of every time.
So if I have f of x is equal to 5 times minus x to the eighth,
plus x to the minus eighth, all of that over to
the fifth power.
If I want to figure out f prime of x I just take the derivative
of this inner function I guess I could call it, so that's
minus 8x to the seventh minus 8x-- because it's just take the
negative 8-- to the minus ninth, times the derivative
of this larger function.
So 5 times 5 is 25 times something to the fourth.
And that something is just going to be this expression
minus x to the eighth plus x to the minus eighth.
And we're done.
You could simplify it.
You could multiply this 25 out and do et cetera, et cetera.
Hopefully this gives you more of an intuition of what the
chain rule is all about, and I'm going to do a lot more
examples in the next couple of presentations as well.
See you soon.