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.