Let's break down and solve the problem step-by-step.

Function provided:

$f(x) = \frac{x^6 + 4}{x^6}$

Task:

1. Find $f'(x)$ using the quotient rule.
2. Simplify the function first, then find $f'(x)$.

Step 1: Using the Quotient Rule

The quotient rule for derivatives is: $\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}$ where $u = x^6 + 4$ and $v = x^6$.

• Derivative of $u$: $u' = 6x^5$
• Derivative of $v$: $v' = 6x^5$

Applying the quotient rule: $f'(x) = \frac{x^6 \cdot 6x^5 - (x^6 + 4) \cdot 6x^5}{(x^6)^2}$

Now, let's simplify this expression.

Step 2: Simplify the Function First

Simplify $f(x)$: $f(x) = \frac{x^6 + 4}{x^6} = 1 + \frac{4}{x^6}$

Now, find the derivative of $f(x) = 1 + 4x^{-6}$: $f'(x) = 0 - 24x^{-7} = -\frac{24}{x^7}$

Matching with the Options:

From the options provided, B correctly represents the quotient rule application: $f'(x) = \frac{(x^6) \cdot (x^6 + 4)' - (x^6 + 4) \cdot (x^6)'}{(x^6)^2}$

Finally, the simplified function is: $f(x) = 1 + \frac{4}{x^6}$

Let me know if you would like further details on any of these steps, or have more questions. Here are 5 relative questions for practice:

1. How do you differentiate functions with exponents?
2. What are some common mistakes when applying the quotient rule?
3. Can you explain when it's better to simplify a function before differentiating?
4. What is the product rule and how is it different from the quotient rule?
5. How can you verify the result of a derivative calculation?

Tip: Always check if a function can be simplified before applying the quotient rule. This often makes the differentiation easier.