We are tasked with finding the derivative $f' \left( \frac{\pi}{4} \right)$, where the function $f(x)$ is given by:

$f(x) = \frac{6x + \cos(2x)}{1 + \sin^2(2x)}$

We will use the quotient rule to differentiate this function. The quotient rule states that if $f(x) = \frac{g(x)}{h(x)}$, then the derivative $f'(x)$ is:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$

Here, $g(x) = 6x + \cos(2x)$ and $h(x) = 1 + \sin^2(2x)$.

Step 1: Differentiate $g(x)$

We have: $g(x) = 6x + \cos(2x)$ Differentiating term by term: $g'(x) = 6 - 2\sin(2x)$

Step 2: Differentiate $h(x)$

We have: $h(x) = 1 + \sin^2(2x)$ Using the chain rule, we get: $h'(x) = 2 \cdot \sin(2x) \cdot \cos(2x) \cdot 2 = 4\sin(2x)\cos(2x)$ Using the double angle identity, $2\sin(2x)\cos(2x) = \sin(4x)$, we simplify: $h'(x) = 2\sin(4x)$

Step 3: Apply the Quotient Rule

Now, applying the quotient rule: $f'(x) = \frac{(6 - 2\sin(2x))(1 + \sin^2(2x)) - (6x + \cos(2x)) \cdot 2\sin(4x)}{(1 + \sin^2(2x))^2}$

Step 4: Evaluate at $x = \frac{\pi}{4}$

We now substitute $x = \frac{\pi}{4}$ into the expressions.

For $g'(x)$: $g' \left( \frac{\pi}{4} \right) = 6 - 2\sin \left( \frac{\pi}{2} \right) = 6 - 2 = 4$

For $h(x)$: $h \left( \frac{\pi}{4} \right) = 1 + \sin^2 \left( \frac{\pi}{2} \right) = 1 + 1^2 = 2$

For $h'(x)$: $h' \left( \frac{\pi}{4} \right) = 2\sin(4 \cdot \frac{\pi}{4}) = 2\sin(\pi) = 0$

Thus, the derivative at $x = \frac{\pi}{4}$ is: $f' \left( \frac{\pi}{4} \right) = \frac{4 \cdot 2 - (6 \cdot \frac{\pi}{4} + \cos \left( \frac{\pi}{2} \right)) \cdot 0}{2^2} = \frac{8}{4} = 2$

So, the correct answer is $\boxed{2}$.

Would you like more details on any of the steps?

Here are 5 related questions:

1. How do you use the chain rule in trigonometric functions?
2. What is the significance of the quotient rule in calculus?
3. How do trigonometric identities simplify derivative calculations?
4. How would this calculation change if the function were altered slightly, like $\cos(2x)$ replaced by $\sin(2x)$?
5. How can you interpret the result geometrically at $x = \frac{\pi}{4}$?

Tip: Always simplify the derivative expression as much as possible before plugging in specific values to avoid calculation errors.