If f of x equals three x
divided by the quantity
two sine x plus cosine x,
we want to find f-prime of negative pi.
Because our function
f of x is the quotient
of two functions, to find
the derivative function,
we'll have to apply the quotient rule,
which is given here below.
Where if function f is in the numerator,
and function g is in the denominator,
the derivative of this
quotient with respect to x
is equal to g times f-prime
minus f times g-prime
divided by g-squared.
Which means, in our case, f-prime of x
will be equal to a quotient.
We'll notice how the denominator is just
a denominator-squared so we would have
the quantity two sine x plus cosine x
squared.
And then our numerator is going to be
g times f-prime minus f times g-prime.
Well again, g is just the denominator,
so we have two sine x plus cosine x
times f-prime which
would be the derivative
of three x, which is just three,
and then we have minus f, which is three x
times the derivative of g, which would be
the derivative of two
sine x plus cosine x.
Well the derivative of two sine x
would be two times cosine x.
The derivative of cosine
x is negative sine x,
so we have minus sine x.
So this would be our derivative function,
and because we're trying to
evaluate this at negative pi,
we're not going to simplify this.
We'll now substitute negative pi for x.
So f-prime of negative pi
would be the same quotient here
with x replaced with negative pi.
So our denominator is going to be
two sine negative pi
plus cosine negative pi.
This is all squared.
The numerator is going to
be two sine negative pi
plus cosine negative pi
times three minus, this would be,
three times negative
pi or negative three pi
times two cosine negative pi
minus sine negative pi.
Now we'll determine the
value of cosine negative pi,
and sine negative pi
using the unit circle.
If we wanted to sketch
negative pi radians,
we would start here and then rotate
clockwise pi radians to here.
So this would be the terminal side
of negative pi radians on the unit circle,
x equals cosine theta
and y equals sine theta.
So cosine negative pi equals negative one
and sine negative pi equals zero.
Since sine negative pi equals zero,
let's go ahead and simplify this.
This would be zero.
This would be zero.
And this would be zero.
And now we'll substitute negative one
for cosine negative pi.
We'd have negative one here times three.
That's negative three.
Next we'd have plus three pi
times two times negative
one, that's negative two.
Now going to the denominator we would have
just negative one squared,
which is positive one.
So this simplifies nicely to
negative three minus six pi.
And just in case we are
asked to round this value,
let's go and get our decimal approximation
to four decimal places.
Just keep in mind, we don't want to round
unless the direction tell us to.
Negative three minus six pi
would be approximately negative 21.8496,
rounded to four decimal places.
This would also be the
slope of the tangent line
at x equals negative pi radians.
I hope you found this helpful.
