We want to determine
the derivative of the given function.
This one looks a little funny
because we have f of x equals
natural log of natural log five x,
so we do have a composite function
so we'll have to apply the chain rule.
And the most important thing here is
to identify the inner function
and then let the inner function equal u
so we can apply our derivative formula
for the derivative of natural
log u with respect to x,
which includes the chain rule.
So for this problem, u is going
to be equal to natural log five x,
so now we can think of this
function as natural log u.
So let's write this out.
We have u equals natural log five x,
so now we also know we're going to have
to determine du, dx or u prime.
But to do this, notice that u
is also a composite function.
Since we've already used u, we can let
this inner function of five x equal v
and the derivative of natural log v
would be one over v times v prime,
so u prime's going to be one over five x
times five, or five over one.
Notice how the fives simplify out,
so u prime is equal to one
over x, or one divided by x.
So now that we now have u
and u prime, we can determine
the derivative of natural
log u with respect to x.
So f prime of x is going to be equal
to one over u times u prime.
So we'll have one over natural
log five x times u prime
and u prime is one over x.
So our derivative function
is going to be equal to
one divided by x times natural log five x.
So as you can see, this
wasn't so bad after all,
as long as we take our time,
identify the inner function,
determine the derivative
of the inner function,
and then apply the derivative
formula for natural log u
which includes the chain rule.
