We want to determine the derivative
of f of x equals the quantity two x
to the fifth minus three
raised to the eighth power.
So we need to recognize that
this is a composite function
where we have an inner
function and an outer function
and, therefore, will have
to apply the chain rule
to find the derivative.
The chain rule states
to find the derivative
of a composite function,
we need to determine the
derivative of the outer function
and then multiply it by the derivative
of the inner function.
But if you look at the
notation for the chain rule,
it can be a little bit overwhelming.
So let's take a look at it.
Once we know we have a composite function,
one of the most important
things you need to do
is identify the inner function
of the composite function
and once we identify the inner function,
we want to let that equal u.
So in this case, notice that
g of x is the inner function
so we let u equal g of x.
Once we know u,
we can rewrite the function in terms of u
and then apply the chain rule
and the chain rule can be
expressed two different ways.
One way using Leibniz notation
and the second way
using function notation.
Using Leibniz notation,
dy dx is equal to dy du times du dx
Well, dy du represents the
derivative of the outer function
and du dx represents a
derivative of the inner function.
Looking at the function notation,
f prime of u is the same as dy du,
which is a derivative
of the outer function
and then u prime is a
derivative of the inner function
which again, is the same as du dx
and then once the chain
rule is introduced,
all the basic derivative
formulas are given again
with the chain rule built in
as we see here below.
So notice how here instead
of x to the power of n
we have u to the power of n.
Again, u being the inner function
and to find the derivative of u to the n,
we find the derivative with respects to u
and then multiply it by u prime
which again, is the derivative
of the outer function times
the derivative of the inner function.
So looking at our function,
we need to recognize that
the inner function would be
two x to the fifth minus three
which we'll let equal u.
Now let's go ahead and
write that out over here.
U is equal to two x to
the fifth minus three.
So if the inner function is u,
we could write this as u to the eighth
and so now we have the
function written in terms of u,
we can apply the chain rule
by applying the general,
or extended power rule which means
to find f prime of x,
we'll determine the derivative of
the function with respects to u.
Well, the derivative of u to the eighth
would be eight u to the seventh
and now we need to multiply
it by the derivative
of the inner function
which would be u prime.
So now we've applied the chain rule,
the only thing left to do is
to rewrite this derivative
in terms of x rather than u.
So instead of u to the seventh,
we'll have the quantity two x to the fifth
minus three to the seventh
and then we'll have to determine u prime.
Well if this is u, u prime
would be ten x to the fourth.
So we have all the information we need
to write this in terms of x.
F prime of x is equal to eight times
the quantity two x to
the fifth minus three.
That's u raised to the
seventh and then u prime
is 10 x to the fourth.
So if we can write this one more time,
f prime of x is equal to,
well, eight times 10 x to
the fourth would be eighty,
x to the fourth times the quantity two x
to the fifth minus three
to the power of seven.
This would be our derivative function.
So even though the first
time you see the chain rule
it can be a bit overwhelming,
as long as you can identify
the inner function as u,
rewrite the function in terms of u,
it is a fairly straightforward process.
We find the derivative of
this with respects to u
and then multiply by u prime.
We'll take a look at more examples
in the next several videos.
I hope you found this helpful.
