We're given f of x
and asked to find the derivative function,
or f prime of x, and f prime of pi divided by six.
Looking at the given function,
notice how we have a quotient.
So if we leave the function in this form,
we'll have to apply the quotient rule given here.
The other option is to write the trig functions
in terms of sines and cosines
and see if we can simplify this function.
Let's go ahead and try to simplify it
before deciding if we have to use the quotient rule.
So in terms of sine and cosine, we can write this as f of x
equals secant x is equal to one over cosine x.
So in parentheses we'd have one divided by cosine x
plus two divided by tangent x,
which is equal to sine x divided by cosine x.
Now, instead of dividing, let's multiply by the reciprocal.
So we'd have the quantity one divided by cosine x
plus two times the reciprocal of sine x over cosine x,
which would be cosine x divided by sine x.
And now we'll distribute here and here.
When multiplying one over cosine x
and cosine x over sine x,
notice cosine x over cosine x is simplified to one.
We'd be left with one over sine x, which equals cosecant x.
And then for the second product, nothing simplifies.
We have two times this fraction here,
but cosine x divided by sine x is equal to cotangent x.
So we have plus two cotangent x.
So this helps us quite a bit,
because now we can find the derivative function
without having to apply the quotient rule given here.
F prime of x is equal to the derivative of cosecant x
plus two times the derivative of cotangent x.
Well, the derivative of cosecant x is equal
to negative cosecant x cotangent x.
And the derivative of cotangent x is equal
to negative cosecant squared x.
So we'd have two times negative cosecant squared x,
or just minus two cosecant squared x.
So this is the first part of the question.
We just found the derivative function.
And now we want to evaluate this at x equals p divided by six,
which would give us a slope of the tangent line
at x equals pi over six.
So f prime of pi over six would be equal
to negative cosecant pi over six
times cotangent pi over six
minus two times cosecant squared of pi over six.
To find these trig function values,
let's go ahead and use 30-60-90 reference triangle,
since pi over six radians is equal to 30 degrees.
So if this angle is 30 degrees
and we have a 30-60-90 right triangle,
we can label the short leg one,
the hypotenuse two, and the long leg square root of three.
And therefore cosecant of pi over six radians
would be equal to the ratio of the hypotenuse
to the opposite side
since cosecant is the reciprocal of sine,
which would be two divided by one or just two.
And cotangent pi over six radians is equal to the ratio
of the adjacent side or the opposite side.
If cotangent's the reciprocal of tangent,
the ratio of the adjacent side to the opposite side
would be square root of three divided by one,
or square root of three.
So f prime of five or six would be equal
to negative cosecant pi over six is two
times cotangent pi over six, which is square root of three,
minus two times cosecant squared pi over six,
where cosecant pi over six is two,
so this would be two squared.
So we have negative two square root of three minus,
this would be four times two, so minus eight.
So this would give us the slope of the tangent line
at x equals pi over six.
Let's also get a decimal approximation for this
to three decimal places.
Negative two times square root of three
minus eight.
Rounding to three decimal places,
this would be approximately negative 11.464.
Let's take a look at the graph of the function
and locate the point when x equals pi over six
and see if the slope of the tangent line does look
like it's approximately negative 11.464.
So here's the graph of our function.
Here's the point on the function
where x is equal to pi over six radians.
The red line is a tangent line at that point.
And since f prime of pi over six
was approximately negative 11.464,
this would be the slope of this red tangent line.
It would also be the instantaneous rate of change
of the function at x equals pi over six.
And notice since it is negative,
the function is also decreasing at that point.
I hope you found this helpful.