We want to find the derivative
of f of x equals four
times e raised to the power
of two sine of three x to the fourth.
The first thing we should recognize
is that the given function
is a composite function,
and therefore, to find the derivative
we'll have to apply the chain rule.
The main idea of the chain rule is to find
the derivative of the composite function,
we find the derivative
of the outer function
and then multiply by the
derivative of the inner function.
Many times, once we learn the chain rule,
we're given derivative formulas
where the chain rule is
built in as we see here below
where u is equal to the inner function
and notice how the derivative is now given
as a product where the first factor
is a derivative of the outer function
and the second factor, or u prime,
is equal to the derivative
of the inner function.
So looking at our function
and this first derivative formula here,
we should recognize that
the inner function u
is going to be equal to two
sine of three x to the fourth.
So if we let u equal two
sine three x to the fourth,
then we can write f of
x as four e to the u
which means f prime of x
is going to be equal to
four times e to the u
times u prime or e to
the power of two sine
of three x to the fourth times u prime
which should be the derivative
of two sine three x to the fourth.
What's more challenging about this example
is that, notice, two sine
of three x to the fourth
is also a composite function.
So we have to apply the chain rule again
in order to find this derivative.
And looking at this function,
we should recognize that
three x to the fourth
is going to be the inner function.
And in this case, since
we've already used u,
let's go ahead and let this equal v.
So now we have f prime of x
equals four times e to the
two sine three x to the fourth
times the derivative of two sine v
which is going to be equal
to cosine v times v prime
or two times cosine three x to the fourth
which is v.
And then v prime is going
to be the derivative
of three x to the fourth which
would be 12 x to the third.
So here's our derivative
which required applying
the chain rule twice.
Let's go ahead and clean this up.
We have four times two
times 12 x to the third.
It's going to be 96 x to the third
e to the power of two
sine three x to the fourth
times cosine three x to the fourth.
This would be our derivative function.
I hope you found this helpful.
