www.mathcentre.ac.uk/.../8.6%20Chain%20Rule.mp4

Sometimes we have functions that
look like this.

Cause of X squared.

Now there are immediate things
that are different about this

than the straightforward cosine
function. It's cause of X

squared, not just cause of X,
and so we call this function

of a function Y equals a
function of another function.

Now in this case.

The function F is the cosine
function, and the function G is

the square function. Or we could
identify them, perhaps a bit

more mathematically by saying
that F of X is cause

X&G of X is X squared.
Now let's have a look at

another example of this.

This time we'll turn it around,
so to speak. Let's have a look

at cause squared X.

Now, how is this a function of a
function? Let's remember what

cost squared of X means. It
means cause X squared cause X

multiplied by itself. So now if
we look at our function of a

function. Let's see what we can
do to identify which is F and

which is G of X. So here we are
squaring, so it's the F is the

square function, and inside the
square function, the thing that

we are actually squaring is G of
X, namely cause X. So here the F

is the square function and the G
is the cosine function. Or write

it down more mathematically F of

X. Is X squared? Angie of
X is cause X?

Now how do we differentiate
a function like Cos squared

X or a function like the
cause of X squared? What we

need to do to be able to
differentiate something that

is a function of a function?

To do that, we need
to do two things. One,

we need to substitute.

You equals G of X. This would
then give us Y equals F of

you, which of course is much
simpler than F of G of X.

Next we need to use a rule
or a formula that's known as the

chain rule. China
rules quite simple,

says DY by DX is
equal to DY by EU

Times du by The X.

Notice it looks as though the D
use cancel out. If these were

fractions, which they're not, it
looks as though they might

cancel out. That's a way of

remembering it. So this is how
we're going to approach it.

Substitute you equals G of X.

And then apply this rule called
the chain rule in order to find

the derivative. So let's take a
number of examples and the first

one will take is the very first
example that we looked at.

And that was why equals
cause of X squared.

So our first step was to
put you equals X squared.

And then Y is
equal to cause you.

Now the chain rule says DY
by the X is equal to

DY by du times du by
the X. So we need.

Do you buy the X?

Well, that's 2X. We
differentiate this. We multiply

by the two and take one off
the index. That leaves us 2X and

we need the why by DU.

And the derivative of Cos you is
minus sign you.

So now we need to put these
together DY by the X is equal

to. Divide by do you minus
sign U times do you buy DX

which is 2 X?

Now this is all very well, but
really we'd like to have

everything in terms of X. And
here we've got you, so we need

to undo this substitution. If we
put you equals X squared, we now

need to replace EU by X squared,
and so we write minus 2X and

bringing the two X to the front
sign of X squared.

Now they're all done like that.
What I'm going to do with all

the next examples that I do is
I'm going to put this

differentiation of the two bits
up here with these two bits.

Let's have a look at the
other one. We had. Y equals

cause squared X. Let's
remember that meant the

cause of X all squared.

So this is now the G
of X and so we will

put you equals cause of X.

Then Y is equal
to you squared.

I can calculate do you
buy DX that's minus sign

X and I can calculate
DY by du. That's two

you. Write down the chain
rule the why by DX is

DY by du times DU by

The X. And now we can substitute
in the bits that we've already

calculated. The why by do you is

to you. Times do you buy DX
which is minus sign X equals.

Now again we want this all in
terms of X. So what we have to

do is reverse the substitution
we put you equals to cause X and

now we need to undo that by
replacing EU with cause X. So

bringing the minus sign to the
front minus 2.

Kohl's X

Sign X.
Let's take

another example.

Y equals 2X minus five
all raised to the power

10. Now, it might be tempting to
say, well surely we could just

multiply out the brackets, but
this is to the power 10 to

multiply out. Those brackets
would take his ages, and there's

all those mistakes that could be
made in doing it. Plus when we

differentiate it, we may not
have the best form for future

work, so let's use function of a
function. So here we will put

you equals this bit here inside

the bracket. 2X minus five

and then. Why is
equal to U to the power

10?

I can now do the differentiation
of the little bits. Do you buy

the X is just two the derivative
of two X just giving us two and

E? Why by DU is 10 we multiply
by the index U to the power 9.

We take one away from the index.
Now we can put this together

using the chain rule so divided
by DX is equal to.

The why by DU times
DU by X, substituting our

little bits. Here's the why
by DU10U to the ninth

times two do you buy

DX? 2 * 10 is 20 and I
want you to the power 9 and I

need to get this all in terms of
X, so I need to replace the you

here. By the two X minus five.
So that's 2X minus five all to

the power 9, and that's a
compact expression for the

derivative. Think what it would
have been like if I had to

expand the brackets and
differentiate each term.

I want now to take another trick
example and then develope that

trig example a little bit
further to a more general case.

So we'll take
Y equals the

sign of 5X.

Very easy here to identify the
G of X. It's this bit

here the 5X so will put
you equals 5X.

And then why will be
equal to sign you?

Differentiating do you buy the
X is equal to 5.

And DY bite U is
equal to cause you.

Put the two bits together with
the chain rule DY by the

X is equal to DY by
EU Times DU by The X.

DIY Bindu is cause you.

Times by and you by the
ex here is 5.

So let's bring the five to
the front 5 cause of, and

now let's reverse the
substitution. You is equal to

5X, so will replace EU by

5X. Now, notice how that five
here and here is apparently

appeared there, and it did so
because the derivative of 5X was

five. So the question is, could
we do this with any number that

appeared there in front of the X
bit five or six? Or 1/2 or

not .5? Or for that matter, an

so? Have a look at
Y equal sign NX.

You equals an X that

sour substitution. And then Y
is equal to sign you we can

differentiate EU with respect to
X and the derivative of NX is

just N because N is a constant

and number. And the why by
DU is equal to cause you.

We can now put this back

together again. My by the X
is DY by EU Times

DU by The X.

Equals. DIY
Bindu is cause you.

Times by du by DX, which is just

N. So let's move the
end to the front. That's

N cause NX.

So the ends of behaved in
exactly the same way that the

fives behaved in the previous
question. And of course this

means. Now we're in a position
to be able to do any question

like this simply by writing down

the answer. So if we're just
going to write down the answer,

let's take. Why is
sign of 6X?

And then divide by the X is
just six cause 6X just by using

the standard result that we had

before. Similarly, if we had
Y was equal to cause 1/2, X

will just because it's cause
it's not going to be any

different to sign really and
so D. Why by DX is equal

to 1/2 and the derivative of
causes minus sign 1/2 X.

Now let's take one more example.
In this particular style, so

will take Y equals E to
the X cubed. And here this

X cubed is our G of
X. So we will put you

equals X cubed.

And then why will be equal 2
E to the power you we can

differentiate. So do you find
the X is equal to three X

squared? We multiply by the
power and take one off it. And

why by DU is equal to E
to the power you we

differentiate with respect to
you. The exponential function is

it's. Own derivative, and so it
stays the same.

Bringing those two bits together
to give us the why by DX through

the chain rule DY by DX is DY by
du times by DU by The X.

The why Bindu is E to the power
you times by du by DX which is

3 X squared.

Move the three X squared through
to the front.

E to the power. And now we made
the substitution that gave us

you we put you equals X cubed
and now we need to.

Reverse that substitution and
replace U by X cubed.

So let's have a look at what
we've got here.

E to the X cubed. Now if we
think of this as E to EU then E

to EU is just the derivative of
the F with respect to you.

Because the exponential function
is own derivative and here the

three X squared is simply the
derivative of U with respect to

X. In other words, it's the
derivative of G of X. Now let's

see if we can put that together
for a general case.

So we've got Y equals
F of G of X.

And we're putting you equals G
of X and our chain rule tells

us that the why by the X
is equal to.

DY by EU Times
DU by The X.

Equals let's start here. You
remember that in most of the

examples that I've done, I've
been pushing the result of du by

DX forward to the front of the

expression. So let's do it to
begin with. Instead, at the end

when we've been simplifying, so
do you buy DX? This is the

derivative of G of X, so that's
DG by D of X by DX.

Times by and this is the why
by DU. So that's the derivative

of F of G of X with
respect to you.

Now can we use this? Let's have
a look at an example why equals

the tan of X squared and so DY
by the X is equal to.

Well, this bit here is the G of
X, so we need its derivative

and we're going to write it
down at the front. That's easy,

that's 2X. And now we need to
differentiate tan as though

this was tamu with respect to
you and the derivative of tan

is just sex squared. And then
we need to put in the G of X

sex squared X squared.

Notice how short this is. 2
lines. Let's just look back at

the example that we did before
and it took us all of this.

What we've done is we've
contracted this whole process to

one that takes place in our

head. Quicker. You can still do
it this way if you like, but if

you can get into the habit of
doing this, it's much quicker.

So let's have a look at some
examples and try and keep track

of what we're doing. So let's
begin with Y equals E to the one

plus X squared.

Sunday why by DX is equal to.

Well, we can identify the G of
X. It's this up here. The power

of the exponential function, and
so if we differentiate that it's

2X. And now we want the
derivative of the exponential

function. It is its own
derivative, so that's E to the

one plus X squared.

Able to write it down straight

away. Let's take Y
equals the sign of X

Plus E to the X.

The why by DX is equal to

well here. We can identify the G
of X, so we want the derivative

of this to write down the
derivative of X is one. The

derivative of E to the X is just
E to the X.

Times by and we want the
derivative of this.

As though this were you and the
derivative of sine is cause, and

then X Plus E to the power X.

We take
Y equals

the tan.
All X squared

plus sign X.

Why buy the X is equal
to identify the G of X?

That's this lump here, the X
squared plus sign X

differentiate it 2X plus cause

X. And now the derivative of
tan as though it were tan of

U. Well that will be sex
squared. But instead of EU

we want you is G of X which
is X squared plus sign X.

Y equals this time will have a
look at one that's got brackets

and powers in it. 2 minus X to
the 5th, all raised to the 9th

power. The why by DX is got
to identify the G of X first and

the G of X is this bit inside
the bracket the two minus X to

the power 5. So we need to
differentiate that first, so

that's going to be minus 5X to
the power 4.

Remember the derivative of two,
which is a constant is 0 and

then multiply by the Five and
take one off the index of minus

5X to the power 4.

Now we've got to differentiate
to the power 9 as though it was

you to the power 9 will that
would be times by 9, and then

instead of EU we want.

The two minus X to the 5th and
we take one away from the

power. We get that little bit
of tidying up to do here,

'cause we've got a
multiplication that we can do

minus five times by 9 - 45 X
to the power four 2 minus X to

the fifth, raised to the power
8.

Take 1 final example.

Let's take Y equals the
log of X plus sign

X. We can identify this as our
G of X.

So D why by DX is equal to
we can write down the derivative

of this one plus cause X times
by and now we want the

derivative of log.

Well, the derivative of log of
you would be one over you, so

this is one over and this is EU
or the G of X one over X plus

sign X and it be untidy to leave
it like that. So we move that

numerator one plus cause X. So
it's more obviously the

numerator of the fraction. So
what we've seen there is the use

of the chain rule.

But then modifying it slightly
so we get used to writing down

the answer more or less
straight away. You can still

use the chain rule. You can
still do it at full length, but

if you can get into the habit
of doing it like this, you find

it quicker and easier to be
able to manage and more

complicated problems will be
easier to do if you can short

cut short circuit some of the
processes.