We're asked
to determine the derivative function,
given f of x equals cosine of the quotient
of 1 plus e to the 2x and 1 minus e to the 3x.
Notice how we have a composite function, and therefore,
we'll have to apply the chain rule to find the derivative,
and also notice how the inner function is
a quotient function, and therefore,
we'll let u equal the quotient function.
And therefore, we can think
of the function f of x as just cosine u
when determining the derivative.
Notice the derivative of cosine u with respect to x is equal
to negative sine u times u-prime.
So, if we let u equal the quotient function.
Notice we do have to find u-prime, or du/dx,
to apply the chain rule.
So, let's work on determining u-prime.
Notice how to find u-prime,
we will have to apply the quotient rule of differentiation.
The denominator of u-prime is just the denominator squared,
or the square of 1 minus e to the power of 3x.
In the numerator, we have denominator
of the quantity 1 minus e to the 3x times the derivative
of the numerator, which is the derivative
of 1 plus e to the 2x.
And then minus the numerator
of the quantity 1 plus e to the 2x times the derivative
of the denominator,
which is the derivative of 1 minus e to the 3x.
Next, we find the derivative here and here,
and then simplify as much as possible.
So, the denominator is going to remain the same.
In the numerator, we have the quantity 1 minus e to the 3x,
and then times the derivative of 1 plus e to the 2x
with respect to x, which is zero plus e
to the power of 2x times the derivative of 2x,
applying the chain rule,
which gives us 2e to the x times 2, or 2e to the 2x.
Then we have minus the quantity 1 plus e
to the 2x times the derivative of 1 minus e to the power
of 3x with respect to x,
which is zero minus e to the power
of 3x times the derivative of 3x,
which is 3 giving us negative 3e to the 3x.
And now, let's go ahead and distribute in the numerator,
and simplify as much as possible.
The denominator remains the same.
Distributing 2e to the 2x, we have 2e to the 2x,
and then 2e to the 2x times negative e to the 3x.
Remember when the bases are the same, we add the exponents.
This gives us minus 2e to the power of 5x.
Notice 3x plus 2x is 5x.
Next, we distribute negative 3e to the 3x,
and then subtract,
negative 3e to the 3x times 1 is just negative 3e to the 3x,
but we are subtracting, and therefore,
we get plus 3e to the 3x.
And for the next product we have negative 3e to the 5x,
but again, we're subtracting,
which gives us plus 3e to the 5x.
Notice how we have two e to the 5x terms,
and therefore, we can combine like terms.
U-prime is equal to negative 2e to the 5x plus 3e
to the 5x is 1e to the 5x, or just e to the 5x.
And then, we have plus e to the 3x plus 2e to the 2x,
all divided by the square of 1 minus e to the 3x.
So again, now we know if we let u equal
the quantity 1 plus e to the 2x divided
by the quantity 1 minus e to the 3x.
U-prime is equal to, again, e to the power of 5x plus 3e
to the power of 3x plus 2e to the power of 2x all divided
by the square of 1 minus e to the 3x.
This is what we need in order to determine f-prime of x.
F-prime of x is equal to the derivative of cosine u
with respect to x,
which again is equal to negative sine u times u-prime,
which gives us negative sine of the quantity 1 plus e
to the 2x divided by the quantity 1 minus e
to the 3x all times u-prime,
which we now know is equal to, again, e to the 5x plus 3e
to the 3x plus 2e to the 2x all divided by the square
of 1 minus e to the 3x.
And we'll go ahead and leave the derivative function
in this form.
I hope you found this helpful.