Hello,
and welcome to the chain rule part one.
The goal of this video will be to use the chain rule
to differentiate a variety of functions.
The point of the chain rule is to allow us
to differentiate composite functions.
This function is a composite function.
You could think of the inner function as two x plus three
and the outer function as a function
that raises something to the seventh power.
Remember in algebra, if we let f of x equal x to the seventh
and g of x equal to two x plus three,
if we were asked to find the composite of f of g,
by definition this is true.
Therefore, we'd start with the inner function, or g of x,
g of x is equal to two x plus three,
and this would be our input into f,
and the result would be the function above,
the quantity two x plus three raised to the seventh power.
So we can use something called the chain rule
to find the derivative of this,
and it will shorten our work dramatically.
Here's what the chain rule says.
Y equals f of u is a differentiable function of u
and u is equal to g of x is a differentiable function of x,
then y equals f of g of x is a differentiable function of x,
and dy dx is equal to dy du times du dx.
Now that's quite a mouthful.
The important part here to remember
is that our u is going to be our inner function.
We're going to rewrite y in terms of u,
differentiate that, and multiply it by the derivative
of u with respects to x.
Now there's a couple ways to write the chain rule.
This notation down here means the exact same thing
as it does above, it's just written in a different way.
G of x is our inner function, and f is our outer function.
Essentially, we're taking the derivative
of the outer function evaluated at the inner function,
times the derivative of the inner function in both cases.
So let's go ahead and see if we can apply this
to the function we were considering earlier,
f of x is equal to the quantity two x plus three
raised to the seventh power.
So we'll do this both ways.
We'll use this notation, then we'll go down here
and use this notation.
Of course, our answers will be the same.
The key to using this notation, which I prefer,
is letting u equal your inner function.
So we can see that two x plus three would be our u.
And if we let u equal two x plus three,
f of u
would equal u to the seventh power.
Now we're ready to apply the chain rule.
So the derivative of the function
with respects to x, or f prime of x,
is equal to the derivative of f with respects to u.
Well, here's our f.
The derivative of f with respects to u would be
seven u to the sixth
times the derivative of u with respects to x.
Well, here's our u.
The derivative of this with respects to x would be two.
There's our derivative, but now we just have to
replace u with two x plus three.
So the derivative of f is going to equal,
now we can multiply the seven times the two to get 14.
U to the sixth would be two x plus three
raised to the sixth power.
Again, this notation down here means exactly the same thing.
If you want to find the derivative of this composite function,
we take the derivative of f, which is our outer function,
and evaluate it at g of x.
Now to me, this is a little bit confusing
until you do quite a few of them.
But essentially, what it's saying is
the derivative of the outer function,
or the function that raises it to the seventh power,
of course, would be seven times some stuff to the sixth.
Well, that stuff is our g of x, or the inner function,
two x plus three.
And then derivative of g of x,
again derivative of the inner function here,
would again be two.
So of course, we get the same answer.
Now, I like the first method
because it does kind of break it down into smaller steps.
This one kind of skips from the original function
all the way to the derivative,
which is fine if you can follow it.
I think it's easier to follow it
if you let u equal your inner function,
at least until you've done quite a few of these.
Now this leads us to something called
the general power rule.
We've already discussed the power rule.
But the general power rule basically includes
the chain rule in the power rule.
If u is a differentiable function of x
and n is a rational number,
then, again, this u could be any function
in terms of x, raised to the nth power,
and it's going to equal the exponent,
times u to the n minus one, times this u prime.
Well, this u prime is going to be our du dx
mentioned in the chain rule.
Or instead of using u, we could leave it
in terms of the function g of x.
Again, these two mean the same thing.
One's in terms of u, one's in terms of g of x.
Let's try a few and see if this makes sense.
First thing we have to do is rewrite this
with a rational exponent.
So we know we can rewrite the square root
as this quantity raised to the 1/2 power.
Again, my first step is going to be to identify the u.
Okay, so our u is equal to two x cubed minus nine,
which makes our y equal to u to the power of 1/2.
Dy dx would be equal to dy du
times du dx.
Remember, another way to write dy dx
would be to just write y prime is equal
to the derivative of the y with respects to u,
this would be 1/2 u to the 1/2 minus one,
or negative 1/2, times du dx.
The derivative of u with respects to x
would be six x squared.
Now we're just left to clean this up.
Our y prime is equal to 1/2 times six, that would be three.
Then we have an x squared.
Now this is a negative exponent,
so I'm going to move this down to the denominator.
So that would be a positive 1/2 exponent in the denominator,
and our u is equal to two x cubed minus nine.
Now this is to the positive 1/2 power,
or I could put it back as the square root.
So notice, after you've found the derivative,
there is still some substitution left,
and then simplifying.
Let's try another.
When you first look at number two,
you might think you're going to have to apply
the quotient rule, but remember,
I can take this quantity and move it.
And if I move it up to the numerator,
this positive three will turn to a negative three.
So I can rewrite this function.
Let's identify our inner function.
Here's our inner function, u.
Our u is equal to two x to the fourth minus one.
Therefore our y would be equal to five u
to the power of negative three.
Applying our chain rule, dy dx or y prime
is equal to dy du,
now here's y,
derivative of this with respects to u would be
negative 15 u to the negative four,
times again, here it is if you've forgot, du dx.
Here's u.
The derivative of u with respects to x
would be eight x to the third.
So we need to clean this up and replace y
with two x to the fourth minus one.
Negative 15 times eight x cubed
would be negative 120 x to the third.
This is raised to the negative fourth power.
I'll move it down
and replace u with two x to the fourth minus one.
And here we have our derivative using the chain rule.
Okay, this ends part one of the chain rule.
Chain rule part two will cover
using the product and quotient rule
mixed in with the chain rule.
Thank you for watching, and I hope that was helpful.