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.
