We want to find the derivative of f of x
equals negative two cosine cubed,
of 5x squared.
The first thing we should recognize
is that our function
is a composite function
and therefor to find the derivative,
we have to apply the chain rule.
The basic idea of the chain rule
is to find the derivative
of a composite function,
we find the derivative
of the outer function
and then multiply by the
derivative of the inner function.
In many times, once we
learn the chain rule,
the differentiation rules
are given as we see here
in terms of u, where the
chain rule is built in,
and u is the inner function.
Notice how the derivatives
are now a product
where the first factor
is the derivative of the outer function
and the second factor, u prime,
is the derivative of the inner function.
So the most important thing
when applying the chain rule
is to identify the inner function.
So looking at f of x,
since we have the cosine function
being raised to the third power,
it might be helpful to rewrite
this in the form f of x,
equals negative two,
then in parenthesis,
cosine, 5x to the second,
raised to the third power.
In this form I think
it's much easier to see
that the inner function
would be cosine, 5x squared,
so this would be our u,
if we're using these formulas here.
Which means if it's helpful,
we can think of this as
negative 2u to the third.
So to find f prime of x,
we can apply the power rule here
with the chain rule built in,
so we're going to
multiply by the exponent,
that would give us negative six,
we're going to keep u the same
so we'd have cosine, 5x to the second,
subtract one from the
exponent, so that's two,
times u prime, which
would be the derivative
of cosine 5x squared.
What's a little more challenging
about this example is that,
notice that cosine and 5x squared
is also a composite function,
where now the inner function is 5x squared
and since we've already used u,
we could let the inner
function be equal to v,
if it's helpful.
So I'll have to apply the chain rule again
in order to find this derivative.
Which means the original function
is actually a composition
of three functions.
So we have f prime of x
equals negative six,
cosine, 5x squared to the second,
times the derivative of cosine, 5x squared
which we could think of as cosine v,
so we'll have negative
sine v times v prime
or negative sine,
5x squared,
times the derivative of 5x squared
which would be 10x.
Now we need to clean this up
so we have f prime of x
equals negative six times 10x,
is negative 60x,
we have a negative here
so it's positive 60x,
and we have two factors
of cosine 5x squared
so I'll write this as cosine squared,
5x squared, one factor
of sine, 5x squared.
So this would be our derivative function
where we applied the chain rule twice
in order to find this derivative.
I hope you found this helpful.
