To determine the derivative
of the given function,
we need to recognize this
as a quotient of two
differentiable functions,
so we'll have to apply the quotient rule
to determine this derivative.
And the quotient rule is listed
down here in red for reference.
So the function in the
numerator will be f,
and the function in the
denominator will be g.
To determine f prime of x,
let's start with the denominator.
Notice how the denominator
is just g squared
or the denominator squared,
so we'll add the quantity
one plus cosine x squared.
Our numerator is going
to be g times f prime
minus f times g prime.
So for g times f prime,
we'll have the denominator,
one plus cosine x,
times the derivative of the numerator.
Derivative of sine x is cosine x,
minus f times g prime, or the numerator
times the derivative of the denominator.
Well, the derivative of one is zero,
derivative of cosine x is negative sine x.
Now that we've applied the quotient rule,
we have our derivative, but
now we have to simplify this
as much as possible.
So we'll leave the denominator the same.
Here we'll distribute cosine x,
so we'll have cosine x
plus cosine squared x.
Here we have minus sine
x times negative sine x,
or minus negative sine squared x,
which becomes plus sine squared x.
Now we should recognize
that cosine squared x
plus sine squared x is equal to one.
So we have f prime of x is equal to,
instead of writing cosine x plus one,
I'm going to write one plus cosine x
all over the quantity one
plus cosine x squared,
which we could rewrite
as one plus cosine x
times one plus cosine x.
So we can see there's a common factor
of one plus cosine x
between the numerator and denominator,
which simplifies to one.
So our derivative function is one
all over one plus cosine x.
And this cannot be simplified any further.
We cannot write this
a sum of two fractions
because we don't have a
monomial in the denominator.
So be careful, this is not equal
to one plus one over cosine x.
This is our derivative in simplest form.
I hope you've found this example helpful.
