https://file.aiquickdraw.com/mathbot/file/17276610221675l09gkgg.jpg

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.

Find f'(x) two ways: (1) using the quotient rule and (2) simplifying first. f(x) = (x^6 + 4) / x^6. Which of the following shows the result of using the quotient rule?

Learn how to differentiate the rational function f(x) = (x^6 + 4) / x^6 using the quotient rule and simplification. This problem helps understand how to apply the quotient rule to find derivatives and the importance of simplifying functions first. Suitable for students in Grades 11-12.

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