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To solve the given problem, we are asked to find the derivative of \( f(x) = \frac{x^6 + 4}{x^6} \) using two methods: 

1. **Quotient Rule**, and 
2. **Simplifying first**.

### Method 1: Using the Quotient Rule

The quotient rule states:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}
\]

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

- \( u(x) = x^6 + 4 \)
- \( v(x) = x^6 \)

Now we compute the derivatives:

- \( u'(x) = 6x^5 \) (since the derivative of \( x^6 \) is \( 6x^5 \), and the derivative of a constant is 0)
- \( v'(x) = 6x^5 \)

Now apply the quotient rule:

\[
f'(x) = \frac{x^6 \cdot 6x^5 - (x^6 + 4) \cdot 6x^5}{(x^6)^2}
\]

Simplify the numerator:

\[
f'(x) = \frac{6x^{11} - 6x^5(x^6 + 4)}{x^{12}} = \frac{6x^{11} - 6x^{11} - 24x^5}{x^{12}}
\]

This simplifies to:

\[
f'(x) = \frac{-24x^5}{x^{12}} = -\frac{24}{x^7}
\]

So, the derivative using the quotient rule is:

\[
f'(x) = -\frac{24}{x^7}
\]

The correct answer from the multiple choices provided would be **option B**, as it matches the quotient rule expression before simplifying.

### Method 2: Simplifying First

We can rewrite \( f(x) = \frac{x^6 + 4}{x^6} \) by splitting it into two terms:

\[
f(x) = \frac{x^6}{x^6} + \frac{4}{x^6} = 1 + 4x^{-6}
\]

Now, differentiate term by term:

- The derivative of \( 1 \) is 0.
- The derivative of \( 4x^{-6} \) is \( -24x^{-7} \).

Thus:

\[
f'(x) = -\frac{24}{x^7}
\]

So, both methods give the same result:

\[
f'(x) = -\frac{24}{x^7}
\]

### Summary:
- Using the quotient rule, we find that \( f'(x) = -\frac{24}{x^7} \).
- Simplifying first gives the same result: \( f'(x) = -\frac{24}{x^7} \).
- The correct answer in the multiple-choice section is **option B**.

Would you like further details or clarifications on any steps?

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Here are 5 related questions:
1. How do you apply the product rule for differentiation?
2. What is the chain rule in calculus, and when do you use it?
3. How can you differentiate a function with multiple variables?
4. What happens when you apply implicit differentiation to an equation?
5. How would you integrate a function like \( x^{-7} \)?

**Tip:** Always try simplifying expressions before applying differentiation rules like the quotient rule to make calculations easier!

Find f'(x) for f(x) = (x^6 + 4) / x^6 using both the quotient rule and simplifying first.

Learn how to differentiate f(x) = (x^6 + 4) / x^6 by using the quotient rule and by simplifying the function first. This problem covers key calculus concepts such as the quotient rule and basic derivative rules. Suitable for high school calculus students.

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